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N THE DOCTRINE O F CHANCES: O R, A Method of Calculating the Probabilities of Events in Play. THE THIRD EDITION, Fuller, Clearer, and more CerreSl than the Former. By A. D E M O I V R E, Fellow of the Royal Society, and Member of the Royal Academies OF Sciences of Berlin and Paris. LONDON: Printed for A. Millar, in the S^ra^d. MDCCLVI, |q n D m Q n a u a u q| To the Right Honourable the Lord CARP ENTER.* MY LORD, HERE are many People in the World who arc prepoffefled with an Opinion, that the Dodrine of Chances has a Ten- dency to promote Play; but they foon will be undeceived, if they think fit to look into the general Defign of this Book : in the mean time it will not be improper to in- form them, that your Lordfhip is pleafed to efpoufe the Patronage of this fecond Edition; which your ftiidl Pro- bity, and the diftinguifhed Charader you bear in the World, would not have permitted, were not theii' Ap- prehenfions altogether groundlefs, * Tliis Dedication was prefixed to the 2d Edition. Your 880G';'li DEDICATION. Your Lordfhip does eafily perceive, that this Dodlrine is fo far from encouraging Play, that it is rather a Guard againft it, by fctting in a clear Light, the Advantages and Difad vantages ofthofe Games wherein Chance is concerned. Befides the Endowments of the Mind which you have in common with Thofe w^hofe natural Talents have been cultivated by the beft Education, you have this particular Happinefs, that you underftand, in an eminent Degree, the Principles of Political Arithmetic, the Nature of our Funds, the National Credit, and its Influence on public Affairs. As one Branch of this ufeful Knowledge extends to the Valuation of Annuities founded on the Contingencies of Life, and that I have made it my particular Care to facili- tate and improve the Rules I have formerly given on that Subjeft ; I flatter mylclf with a favourable Acceptance of what is now, with the gr^atefl: Deference, fubmitted to your Judgment, by, My Lord, Your Lord/hlfs Moft Obedient and Mojl Obliged, Humble Servant, A. de Moivre. J %>:# %*^ %I# ir» %I# %*a^ %*# I ^ -"a^cf* *a*d?* ns'^tf' «^* ~i-^tf' -a^-cT' -;>-^<j- ofa (*ocJoo&oi!S3o*c5ooSoo5ooto PREFACE 5F~»~^75 wo TO ^^flw/ ^^ip^w 2>^n, fince I gave a SpecimeJi in the I Philofophical Tranladtions, of what I now more largely I treat of in this Book. I'be occafion of my then undertaking -^^ this SubjeSi was chiejly oicitig to the Defire and Rncourage- ment of the Honourable 'f- Francis Robartes Efq; ivho, upon occapon of a French Tra5i, called, L'Analyl'e des Jcux de Hazard, ivhich had lately been publijhed^ ivas pleafed to propofe to me fame Problems of much greater difficulty than any he had Jound in that Book ; which having fohed to his SatisfaStion, he engaged me to methodize thofe Problems, and to lay down the Rules which had led me to their Solution. After I had pro- ceeded thus far, it was enjoined me by the Royal Society, to communicate to them what I had dif covered on this SubjeSl : a fid thereupon it was or- dered to be publijhed in the TranJaSiiotis, not fo much as a matter relating to Play, but as containing fome gefieral Speculations not un-xorthy to be confidered by the hovers of Truth. I had not at that time read any thing concerning this SubjcSi, but Mr. Huygen'i Book de Ratiociniis in Ludo Ales, atjd a Utile E?igliJ}} Piece (which was properly a Tranfation oj it) done by a very ingenious Gentleman, who, tho' capable of carrying the tnatter a great deal far- ther, was contented to follow his Original ; adding only to it the com- putation of the Advajjtage of the Setter in the Play called Hazard, and fome few things more. As for the French Book, I had run it over but curforily, by reafon I had obferved that the Author chiefly injijied on * This Preface was wrttttn in 1717. + Now Earl o/KADtiOR. the ii PREFACE. the Method cf Huygens, ivhich I was abfolutely rcfohed to rejeSi^ as not feeming to me to be the geriuine and natural ivay ^f coming at the So- lution of Prcblems of this kind. I had [aid in my Specimen, that Mr. Huygens iioas the Jirft who had publifjed the Rules of this Calculation, i>2t ending thereby io do jufiice to a Man ivho had ivell deferved cf the Public; but what I then faid lias mi/ivterpreted, as if I had dejigned to urong fome Perfons who bad ccnfdered this matter before him : and a Pafjage was cited againjl me out of Huygcn'i Preface, in which he faith. Sciendum vero quod iam pridem, inter Praeftantiflimos tota Gallia Geometras, Calculus hie fuerit agitatus ; ne quis indebitam mihi primae Invcntionis glo- riani hac in re tribuat. But what follows immediately after, had it been minded, might have cleared me from any Sufpicion oj injuftice. 'The words are thefe, Casterum illi difficillimis quibufque Quseftionibus fe invicem exercere foliti, methodum fuam quifque occultam reti- nuere, adeo ut a prim is dementis hanc materiam evolvere mihi necefie fuerit. By which it appears, that tho' Mr. Huygens was not the fifl who had applied himfelfto tkofe forts of ^efiions, he was ne- "jerthelefs the firfi who had publified Rules for their Solution ; which is all that I ajirmed. ^ Such a 7ra6l as this is may be ufeful to feveral ends; the firfl of which is, that there being in the World feveral inquifitive Perfons, who are defirous to know what foundation they go upon, when they engage in Play, whether from a motive of Gain, or barely Diver- jion, thy may, by the help of this or the like TraB, gratijy their CU' riofity, either by taking the pains to underftand what is here Demofi- ftrated, or el/e making ife of the Conclufions, and taking it for granted that the Demonftrations are right. Another ufe to be made of this DoSirim of Chances is, that it may Jerve i): Conjun^ion with the other farts oJ the Mathematicks, as a ft Introduction to the Art of Rcafoning; it being known by experience that nothing can contribute more to the attaining of that Art, than the con [deration of a long Train of Confequences, rightly deduced from undoubted Principles, of which this Book affords many Examples. To this may be added, that fome of the Problems about Chance having a great appearance of Simplicity, the Mind is eafdy drawn into a belief, that their Solution may be attained by the meer Strength of natural good Se?fe; which generally proving otherwife, and the Mif- takes occafioned thereby being not ui frequent, 'tis pre fumed that a Book of this Kind, which teaches to djftinguifh Truth from what feems fo nearly to refemble it, will be looked upon as a help to good Rea- Jbning. Among PREFACE. iii Among the fe^oeral Mijlakes that are committed about Chance, one of the mojl common and leaft JufpeSied^ is that "which relates to Lotteries. ^hus, Juppofing a Lottery wherein the proportion of the Blanks to the Prizes is as fiiie to one ; 'tis very natural to conclude, that therefore five Tickets are requifite for the Chance of a Prize ; and yet it may be proved, Demofijlratively, that four Tickets are more than Jufficient for that pur- pofe, -which will be confirmed by often repeated Experience. In the like manner, fuppofmg a Lottery nvherein the proportion of the Blanks to the Prizes is as Thirty -nine to One, (fuch as was the L',ttery of 1710^ it may be proved, that in twenty eight Tickets, a Prize is as likely to be taken as mt ; iihich tho' it mayfeem to contradiB the common NutionSy is mverthelefi grounded upon injallible Demcnffration. When the Play of the Royal Oak was in ufe, fome Perfons who lojl confiderably by it, had their Lcjjes chiefly occafioned by an Argument of nifnch they could 7iot perceive the Fallacy. The Odds againfi any par- ticular Point of the Ball were One and Thirty to One, which intitled the Adventurers, in cafe they were winners, to have thirty two Stakes returned, including their own ; injlead of which they having but Eight and Twenty, it was very plain that on the /ingle account of the di fad- vantage of the Play, they loji one eighth part of all the Money they ' layed for. But the Mafler of the Ball maintained that they had no reafon to complain; fince he woidd undertake that any particular point of the Ball f}.^ould come up in Two and Twenty Throws -, of this he would offer to lay a Wager, and a6lually laid it when required. The feeming contradiBion between the Odds of One and Thirty to One, and Twenty-two Throws Jor any Chance to come up, fo perplexed the Adventurers, that they begun to think the Advantage was on their fide ; for which reafon they played on and continued to Ife. I The Doftrine of Chances may likewife be a help to cure a Kind of Superftition, which has been of long ftanding in the World, viz. that there is in Play fuch a thing as Luck, good or bad. I own there are a great many judicious people, who without any other Ajfiftance than that of their 07vn reafon, arefatisfied, that the Notion ofLnck is meerly Chimerical; yet I conceive that the ground they have to look upon it as fuch, may fiill be farther inforced from fome of the following Confiderations. ff by faying that a Man has had good Luck, nothing more was meant than that he has been generally a Gainer at play, the Expreffion might be allowed as very proper in a fhort way of f peaking : But if the Word Good Luck be underflood to fignify a certain predominant quality, fo in- herent in a Man, that he muff win whe?iever he Plays, or at leafi win oftner than lofe^ it may be denied that there is any fuch thing in nature. The iv PREFACE. The AJjerten of Luck are very Jure from their own ExperiencCy that at Jbme times they have been very Lucky, and that at other times they have had a prodigious Run of ill Luck againjl them, which whiljl it continued cbliged them to be very cautious in engaging with the For- tunate ; but how Chance fiouid produce thofe extroardinary Events, is what they cannot conceive : They would be glad, for Injlance, to be Sa- tisfied, hovj they could loj'e Fifteen Games together at Piquet, if ill Luck had not flrangely prevailed againjl them. But if they will be f. leafed to confder tie Rules delivered in this Book, they will fee, flat though the Cdds again/l their lofing Jo many times together be very great, viz. 32767 ^o i, yet that the Pofjibility oj it is 7iot dejlroyed by the greatnejs of the Odds, there being One Chance in 32768 that it may fo happen ; from whence it J'ollows, that it was ftill pojjible to come topafs without the Intervention of what they callV^ Luck. Befdes, This Accident of lofing Fifteen times together at Piquet, is no more to be imputed to ill Luck, than the Winning with one fingle Ticket the highejl Prize, in a Lottery of 32768 Tickets, is to be im- puted to good Luck, fince the Chances in both Cafes are perfeSlly equal. But if it be Jaid that Luck has been concerned in this latter Cafe, the Anfwer will be eafy ; for let us fuppoje Luck not exifting, or at leaji let us fuppofe its Influence to be J'uJ'pended, yet the highejl Prize muftjall into fome Hand or other, not by Luck, (for by the Hypothefis that has been laid a fide) but from the rneer nccefjity of its falling fomewhere. Thofe who, contend Jor Luck, may, if they pleaj'e, alledge other Cafes at Play, much more unlikely to happen than the Winning or Lofing fifteen Games togeti.er, yet Jlill their Opinion will never receive any Addition of Strength J romjuch Suppofitions : For, by the Rules of Chance, a time may be computed, in which thofe Cafes may as probably happen as not ; nay, not only fc, but a time may be computed in which there may be any proportion of Odds for their fo happening. But fiippofing that Gain and Lofs were Jo fliSuating, as always to be dijlributed cjually, whereby Luck liould certainly be annihilated; would it be reafonable in this Cafe to attribute the Events of Play to Chance alone? 1 think, onthe contrary, it would be quite otherwife, for then there •w'.uld be more reaj'on to JuJfeB that fome unaccountable Fatality did rule in it : Thus, if two Perfbns play at Crofs and Pile, and Chance alone be fuppcfed to be concerned in regulating the fall of the Piece, is it pro- bable that there jhould be an Equality of Heads and CroJJes? It is Five to Three that in Jour times there will be an inequality ; 'tis Eleven to Five infix, 93 to 35 in Eight, and about 12/01 in a hundred times : Wherefore Chance alone by its Nature conflitutes the Inequalities of Play ^ and there is no need to have recourfe to Luck to explain them. Further, PREFACE. • V Further, the fame Arguments lahich explode the Notion oJLuck, may^ on the other fide, be ujeful in fome Cafes to eftabliJI) a due com- part fen betijoeen Chance and Defign : JVe may imagine Chance and De- fign to be, as it urre, in Competition ivith each other, for the produclicn of fome forts of E'-cents, and may calculate what Probability there is, that thoje Event sfiould be rather owing to one than to the other. To give a Jamiliar Inflance of this. Let us fuppofe that two Packs of Piquet- Cards being fent for, it fJ:ould be perceived that there is, from Top to Bottom, the fame Dtfpofttion of the Cards in both Packs; let us like- wife fuppofe that, fome doubt arifing about this Difpoftion of the Cards, it fhould be quejiioned whether it ought to be attributed to Chance, or to the Maker's Defign : In this Cafe the DoBrine of Combinations de- cides the ^efiion; fmce it may be proved by its Rules, that there are the Odds of above 263 130830000 Millions of Millions of Mi Hi 'Ms of Millions to One, that the Cards were deftgnedly fct in the Order in which they were found. From this lafl Confideration we may learn, in may Cafes, hew to dijlinguifh the Events which are the effeSi of Chance, from thofe which are produced by Defign : The very Doctrine that finds Chance where it really is, being able to prove by a gradual Increafe of Probability, till it arrive at Demonflration, that where Uniformity, Order and Con- fiancy refide, there alfo re fide Choice and Defign. Laflly, One of the principal Ufes to which this Doftrine of Chances may be applied, is the difcovering of fome Truths, which cannot fail of pleafmg the Mind^ by their Generality and Simplicity ; the admirable Connexion of its Confequences will increafe the Pleafure of the Difcovery, and the feeming Paradoxes wherewith it abounds, will afford ver\ great matter of Surprize and Entertainment to the Inquifitive. A very re- markable Inflance of this ?iature may be feen in the prodigious Advan- tage which the repetition of Odds will amount to ; Thus, Suppofing I play with an Adverfary who allows me the Odds of 43 to 40, and agrees with me to play till 100 Stakes are won or lofl on either fide, on condi- tion that I give him an Equivalent for the Gain I am intitled to by the Advantage of my Odds ; the Sfueflion is, what I am to give him, onfuppofing we play a Guinea a Stake: The Anfwer is 99 Guineas and above 1 8 Shillings *, which will feem aimoft incredible, confuierw^ the fmallnefs of the Odds of ^7, to 40. Now let the Odds be in 'any Propor- tion given, and let the Number of Stakes be played for be never fo great, yet one general Conclufion will include all the pojfible Ca/es, and the ap- plication of it to Numbers may be wrought in lefs than a Minute's time. * Guineas ivire then at 2 \P'- 6 ''• ^ / h«ve vi • PREFACE. / have explained, in my Introdudlion to the following Trtatife^ the chief Rules on which the whole Art of Chances depends ; I have done it in the plainefl manner that I could think of to the end it might be (as much as fofjible) of general life. I fatter my felf that thofe who are acntiaintcd with Arithmetical Operations, will, by the help of the In- trodiidion alone, he able to folve a great Variety of ^leftions depending on Chance : I li^ijh, for the Jake of fome Gentlennn who have beenpleajed to Jubfcribc to the printing of msBooh, that I cotddevery where have been as plain as in the Introdud^ion ; but this was hardly praSlJcaMe, t lie In- vention of the great ef part of the Rules being intirely owing to Algebra j yet I have, as much aspo[}ible., endeavoured to deduce from the Algebraical Calculation f'cveral practical Rules, the 'Truth of which may be depend- ed upon, and which tiuiy be very iif'eful to thoje who have contented them~ Jelves to learn only common Arithmetick. On this cccafton, I n^itjl take notice to fuch of my Readers as are well verf'd in Vidgar Arithmetick, that it would not be difficult for them to make themfelves Mafters, not only of all the practical Rules in this Book, but alfo of more ifeful Difcoveries, if they would take the fmall Fains of being acquainted with the bare Notation oj Algebra, which might be done in the hu7idredth part of the Time that isfpent in learning to write Short-hand. One of the principal Methods I have made ufe of in the following Treat ife, has been the DoBrine of Combinations, taken in a Senfe fomewhat more exteifivc, than as it is commonly underjlood. The No- tion of Combinations being Jo well fitted to the Calculation of Chance^ that it naturally enters the Mind whenever an Attempt is made to- wards the Solution of any Problem of that kind. It was this that led me in courfe to the Confideration of the Degrees of Skill in the Adven- turers at Play, and I have made ufe of it in mojl parts of this Book^ as one of the Data that enter the ^leflion; it being fo jar from, per- plexing the Calculation, that on the contrary it is rather a Help aftd an Ornament to it : It is true, that this Degree of Skill is not to be known any other way than from Obfcrvatian ; but if the fame Obfervatixm. conjlantly recur, 'tis Jlrongly to be prefumed that a near EJiimation of it may be made : However, to make the Calculation more precife, and to avoid caufing any needlefs Scruples to thofe who love Geometrical ExaSlnefs, it will he eajy, in the rcom of the word. Skill, to fubflitute a Greater or Lefs Proportion of Chances among the Adventurers, fo as each of them may befaid to have a certain Number of Chances to win one fingle Game^ The general Theorem invented by Sir Ifaac Newton, /cr raifmg a Bino- mial to any Power given, facilitates infinitely the Method of (Zom\>miii\oni,. reprefenting PREFACE. vii reprefenting in one Vieiv the Combination of all the Chances, that can happen in any given Number of Times. 'Tis by the help of that Theorem, joined ivith fome other Methods, that 1 have been able to find practical Rules for the Joking a great Variety of difficult ^te/lions, and to te- duce the Difficulty to a Jingle Arithmetical Multiplication, -whereof fever al In fiances may be Jeen in the /sjbth Page of this Book. yfnot her Method I have made u/e of is that of Infinite Ssrks, •which in many cafes -will folve the Problems of Chance more naturally than Combinations. To give the Reader a Notion of this, we may fiippofe two Men at Play throwing a Die, each in their Turns, and tL.t he is to be reputed the lVi7iner who Jhall frjl throw an Ace: It is plain, that the Solution of this Problem cannot fo properly be reduced to Com- binations, 'which ferve chiefly to determine the proportion of Chances be- tween the Gamejlers, without any regard to the Priority of Play. 'Tis convenient thercjore to have reccurfe to fome other Method, Juch as the following: Let us fuppofe that the firfl Man, being willing to compound with his Adverfary for the Advantage he is ifititled tojrom his firjl Throw, pould ajk him what Confideration he would alloiv to yield it to him ; // may naturally be /iippofed that the Anfwer would be one Sixth part of the Stake, there being but Five to One againfi him, and that this Allowance would be thought a jiijl Equivalent Jor yielding his Throw. Let us likewije fuppofe the fecond Man to require in his Turn to have onefxth part of the remaining Stake for the Confideration oj his Throw ; which being granted, and the firjl Man's Right returning in courje, 'he may claim again one fixth part of the Remainder, and fo on alter- nately, till the whole Stake be exhaufted: But this not being to be done till after an infinite number oj- Shares be thus taken on both Sides, it belongs to the Method o/" Infinite Series to affign to each Man what pro- portion of the Stake he ought to take at firjl, fo as to anfwer exatlly that famous Divifon of the Stake in infinitum ; by means of which ti will be found, that the Stake ought to be divided between the contend- ing Parties into two parts, reJpeSiively proportional to the two Numbers 6 and 5. By the like Method it would be found that if there were Three or more Adventurers playing on the conditions above defcribcd, each Man, according to the Situation he is in with refpecl to Priority of Play, might take as his due fucb part of the Stake, as is exprefjible by the correjponding Term of the proportion of 6 to 5, continued to )d many Terms as there are Gamefiers ; which in the caje of Three Gamefters, for Inflame, would be the Numbers 6, 5, and 4^ , or their Pro\ portionak 16, 30, ,7^25. Another Advantage oJ the Method of Infinite Series is, that every lerm of the Series includes jlme particular Circumftance wherein the ^ 2 Gamefiers viii PREFACE. Gatnejhrs way be founds 'ivhich the other Methods do not ; and that a fe^v of its Steps are Jitfjicient to di /cover the Law of its ProceJ's. The only 'Difficulty v:hich attends this Method, being that of fiimming up fo many of its Terms as are requifite for the Solution of the Problem propofcd: But it will be found by Experience, that in the Series refult- ing from the Confide rat ion cf mojl Cafes relating to Chance, the Therms of it will either ccfjfliiute a Geometric Progrefjion, which by the known Methods is eafily }ummablc ; or elfe f'ome other fort of Progreffion, whofe nature confifis in this, that every Term of it has to a determinate num- ber of the preceding Terms, each being taken in order, fome conflant re~ laticn ; in which cafe I have contrived fome eafy Theorems, not only for finding the Law of that Relation, but alfo for finding the Sums re- quired'-^ as may be feen in fever al places of this Book, but particu- larly from page 2 20 to page 230. A Third /Advantage of the Method of Infinite Series is, that tlje So- lutions derived from it have a certairi Generality and Elegance, which fcarce any other Method can attain to ; thofe Methods being always perplexed with various unknown ^lantities, and the Solutions obtained by them terminating commonly in particular Cafes. There are other Sorts of Series, which tho not properly infinite, yet are called Series, from the Regularity of the Terms whereof they are compofed; thofe Terms following one another with a certain uniformity y which is always to be defined. Of this nature is the Theorem given by Sir Ifaac Newton, in the fifth Lemma of the third Book of his Principles, for drawing a Curve through any given number of Points ; of which the Demonfiration, as well as of other things belonging to the fame Sub]e£i, may be deduced from the firfi Propofition oj his Methodus Differentialis, printed with fotne other of his TroSis, by the care of my In- timate Friend, and very fJiiljul Mathematician, Mr. W. Jones. The abovementio?ied Theorem being very ifeful in fumming up any 7iumber of Terms whofe lafl Differences are equal, (fuch as are the Numbers called Triangular, Pyramidal, &c. the Squares, the Cubes, or other Po%vers of Numbers in Arithmetic Progreffion) I have Jhewn in many places of this Book how it might be applicable to thefe Cafes. After having dwelt Jome time upon various ^uefiions depending on the general Priyiciple of Combinations, as laid down in my IntroduSiion, and upon f'ome others depending en the Method of hfinite Series, 1 pro- ceed to treat of the Method of Combinations properly fo called, which I fi:ew to be eafily deducible from that more general Principle which bad been before explained: JVfjere it may be obferved, that although the Cafes it is applied to are particular, yet the Way of Reafoning; and the Confequences derived from it, are general; that Method of Argiiing PREFACE. i% jirguing about generals by particular Examples, being in my opinion •very convenient jor eafing the Reader's Imagination. Having explained the common Rules of Combinations, and given a The- orem lohich may be of ufefor the Solution offome Problems relating to that EubjeSl, I lay down a new Theorem, which is properly a contraSfion of the former, whereby feveral ^eftions of Chance are refolved with won- derful eafe, tho the Solution might feem at firfl fight to be of infuperabk difficulty. It is by the Help oj that Tloeorem fo contrasted, that I have been able to give a compleat Solution of the Problems o/"Pharaon and Baflette, which was never done before me: 1 own that fome great Mathematicians had already taken the pains of calculating the advantage of the Banker, in any circumjiance either of Cards remairiing in his Hands, or cf any number of times that the Card cf the Ponte is contained in the Stock: But fill the curiofity of the Inqui/itive remained unjatisfed; The Chief ^ejiion, and by much the mojl difficult, concerning Pharaon or Baflette,. being. What it is that the Banker gets per Cent, of all the Money adven- tured at thofe Games ? which now I can certainly anjwer is very near Three per Cent, at Pharaon, and three fcurths per Cent, at Baflette, as may he feen in my 'i^^d Problem, where the precife Advantage is calculated. In the 35^/6 and 26th Problems, I explain a new fort of Algebra, whereby Jome ^ejlions relating to Combinations are jolved by fo eajy a Prccefs, that their Solution is made in fome meafure an immediate con- fequence of the Method of Notation. I will not pretend to fay that this new Algebra is abjolutely necefjary to the Solving of thofe ^cjiions which I make to depend on it, fince it appears that Mr. Monmort, Author of the An^Xyic desjeux de Hazard, and Mr. Nicholas BtrnonWitave folved., by another Method, jnany of the cafes therein propofed : But I hope I J]:all not be thought guilty of too much Confidence, if I affure the Reader, that the Method I have followed has a degree of Simplicity, not to fax of Generality, which will hardly be attained by any other Steps than by. thofe I have taken. The T^gth Problem, propofed to me, amojigft fome others, by the Ho- nourable Mr. Francis Robartes, / had folved in my tradl De menfura Sortis; // relates, as well as the 'i^e^th and ^tth, to the Method of Com- binations, as is made to depend on the fame Principle. When I be^an jor the firfl time to attempt its Solution, I had Jicthing elfe to guide mc but the common Rules of Combinations, fuch as they had been delivered b\ Z)r. Wall is <zff^ others; which when I endeavoured to apply, I was fur- prized to find that my Calculation fwelled by degrees to an intolerable Bulk: For this reafon I was forced to turn my Views another way, and to try whether. X PREFACE. 'whether the Solution I ivas fee king for might not be deduced from Jome ea- fier co'fideratious •■> 'whereupon I happily Jell upon the Method I have beat tnentioning, lihich as it led me to a very great Simplicity in the Solution^ fo I look upon it to be an Improvement made to the Method of Combinations. The Aoth Problem is the reverfe of the preceding ; It contains a very remarkable Method of Solution, the Artifice of ithich conffls in changing an Arithmetic Progrefjion of Numbers into a Geometric one ; this being al'ways to be done ii hen the Numbers are large, and their Intervals fmall. I freely achwivledge that I have been indebted long ago for this ufeful Idea, io my much refpcBed Friend, 'That Excellent Mathematician Dr. Hal- ley, Secretary to the Royal Society, whom I have feen pra5life the thing en another ■ occafion : For this and other Liflruciive Notions readily im- parted to me, during an uninterrupted Friendf}:)ip of five andTwenty years, I return him my very hearty Thanks. The ^^ihand 4 ^th Problems, having in them a Mixture of the two Methods of Combinations and lnf?2ite Series, may be propofed for a pattern of Solution, infome of the moft difficult cafes that may occur in the SubjeB of Chance, and on this occafion I muji do that Jujlice to Mr. Nicholas Bernoulli, to ownhehad fent me the Solution of thofe Problems before mine was Pub- liJJoed ; which I had no fooner received, but I communicated it to the Royal Society, and reprefented it as a Performance highly to be commend- ed: Whereupon the Society order d that his Solution pould be Printed-, which was accordingly donefome time after in the Philofophical Tranf- adions, Numb. 341, where mine was alfo inferted. The Problems which folloiv relate chiefly to the Duration of Play, or to the Method of determining what number of Games may probably be played out by two Adverfaries, before a certain number of Stakes agreed on be- tween them be won or lofi on either fide. This Subject affording a very trreat Variety of Curious ^eflions, of which every one has a degree of '^Difficulty peculiar to itfelf, I thought it neceffary to divide it into fever al diJiinSl Problems, and to illufirate their Solution with proper Examples. Tho' thefe ^eflions may at frfl fight fecm to have a very great degree of difficulty, yet I have fome reafon to believe, that the Steps I have taken to come at their Solution, will eafily be followed by thofe w^ho have a com- petent fki II in Algebra, and that the chief Method of proceeding therein will be undcrficod by thcfe who are barely acquainted with the Elements of that Art. When Ifirfi began to attempt the general Solution oftlje Problem con- cerning the Duration of Play, there was nothing extant that could give me any light into that SubjeB ; for althd Mr. de Monmort, in thefirfi Edition of his Book, gives the Solution of this Problem, as limited to three Stakes to be 'won or lojl, and farther lijmtcd by tl:^e Suppofition of an E- quality PREFACE. xi quality of Skill between the Adventurers ; yet he having given no De- tnonflration of his Solution, and the Demonjlration -when difcovered being of very little ufe towards obtaining the general Solution of the Problem, I was forced to try what my own Enquiry would lead me to which having been attended with Succefs, the refult of what I found was afterwards publifloed in my Specimen before mentioned. All the Problems which in my Specimen related to the Duration of Play, have been kept entire in thejollowing Treatife j but the Method of Solution has received fome Improvemetits by the 7iew Difcoveries I have made concerning the Nature of thofe Series which refult from the Confi- deration of the SubjeB ; however, the Principles of that Method having been laid down in my Specimen, / had nothing now to do, but to draw the Confequerices that were naturally deducible from thetn. ADVERTISEMENT. TH E Author of this Work, by the failure of his Eye-fight in ex- treme old age, was obliged to entruft the Care of a new Edi- tion of it to one of his Friends; to whom he gave a Copy of the for- mer, with fome marginal Corredlions and Additions, in his own hand writing. To thefe the Editor has added a few more, where they were thought neceflary : and has difpofed the whole in better Order; by reftoring to their proper places Ibme things that had been acciden- tally mifplaced, and by putting all the Problems concerning Annuities together; as they fland in the late z/«/>row^ Edition of the Treatife on that Subjedt. An Appendix of feveral ufeful Articles is likewife fub- joined : the whole according to a Plan concerted with the Author^ above a year before his death. E R R A r A. Pag. 10 1. ult. for 445 read 455. p. 27 1. 4, for 10"' read 12?'' Article, p. 29 1. 30, fotxy ready's, p. 45 1. ult. for — s' read ^ x.\ p. 68. 1, 6, for d^ -\- c^ read d^-\- d^. p. 116 1. 26, for Art. %* read Art. 4"". p. 179 1. 8 from the bottom, for XV. Prob. read Car*/, to Prob. 19. p. i8i 1. i, for 18"" read I9'^ p. 187 1. 25, for 38 read 28. p. 192 1. 7, from bottom, for aab and bba readaia and bab, p. 205 1. 7, for of tf I I .-. read . . p. 238. 1. i6, fox AFGz read AFz. THE J.MyndtJiu^- THE DOCTRINE O F CHANGES. Aft«r*J5!*S!f«ft***A*ftft*ft*****ft********************** The INTRODUCTION. H E Probability of an Event is greater or lefs, according to the number of Chances by which , it may happen, compared with the whole num- ber of Chances by which it may either happen or fail. 2. Wherefore, if we conftitute a Fradlion whereof the Numerator be the number of Chances whereby an Event may happen, and the Denominator the number of all the Chances whereby it may either happen or fail, that Fradtion will be a proper defignation of the Pro- B bability / 2 The Doctrine <?/* Chances. bability of happening. Thus if an Event has 3 Chances to happen, and 2 to fail, the Fradlion — will fitly reprefent the Probability of its happening, and may be taken to be the meafure of it. The fame thing may be faid of the Probability of failing, which will likewife be meafured by a Fradion whofe Numerator is the number of Chances whereby it may fail, and the Denominator the whole number of Chances, both for its happening and failing ; thus the Probability of the failing of that Event which has 2 Chances to fail and 3 to happen will be meafured by the Fradtion — . 3. The Fradions which reprefent the Probabilities of happening and failing, being added together, their Sum will always be equal to Unity, fince the Sum of their Numerators will be equal to their common Denominator : now it being a certainty that an Event will either happen or fail, it follows that Certainty, which may be con- ceived under the notion of an infinitely great degree of Probability, is fitly reprefented by Unity. Thefc things will eafily be apprehended, if it be confidered, that the word Probability includes a double Idea ; firft, of the number of Chances whereby an Event may happen ; fecondly, of the number of Chances whereby it may either happen or fail. If I fay that I have three Chances to win any Sum of Money, it Is impoffible from that bare affertion to judge whether I am like to obtain it ; but if I add that the number of Chances either to obtain it, or to mifs it, is five in all, from hence will enfue a comparifon between the Chances that favour me, and the whole number of Chances that are for or againfl me, whereby a true judgment will be formed of my Probability of fuccefs : from whence it neceflarily follows, that it is the comparative magnitude of the number of Chances to happen, in relpedt to the whole number of Chances either to happen or to fail, which is the true meafure of Proba- bility. 4. If upon the happening of an Event, I be intitled to a Sum of Money, my Expedation of obtaining that Sum has a determinate, value before the happening of the Event. Thus, if I am to have lo ^- in cafe of the happening of an Event which has an equal Probability of happening and failing, my Ex- pedlation before the happening of the Event is worth 5 ' .- for I am precifely in the fame circumftances as he who at an equal Play ven- tures 5 ^ either to have 10, or to lofe his 5. Now he who ventures 5 ^- at an equal Play, is pofleflbr of 5 ^- before the decifion of the Plays I'he Doctrine o/*Chances. 3 Play ; therefore my Expedation in the cafe above-mentioned muft alfo be worth 5 ^ 5. In all cafes, the Expeftation of obtaining any Sum is eftimated by multiplying the value of the Sum expected by the Fraction which reprefents the Probability of obtaining it. Thus, if I have 3 Chances in 5 to obtain 100 ^- I fay that the prefent value of my Expedation is the product of 100 ^- by the frac- tion — , and confequently that my expedlation is worth 60 ^• For fuppofing that an Event may equally happen to any one of 5 different Perfons, and that the Perfon to whom it happens fliould in confequence of it obtain the Sum of loo'^- it is plain that the right which each of them in particular has upon the Sum exptded is — — of 100^- which right is founded in this, that if the five Per- fons concerned in the happening of the Event, fliould agree not to fland the Chance of it, but to divide the Sum expeded among them- felves, then each of them muft have — of 100 ^- for his prcten- fion. Now whether they agree to divide that fum equally among themfelves, or rather chufe to ftand the Chance of the Event, no one has thereby any advantage or difadvantage, fince they are all upon an equal foot, and confequently each Perfon's expedation is worth -T- of loo^- Let us fuppofe farther, that two of the five Perfons concerned in the happening of the Event, fliould be willing to refign their Chance to one of the other three j then the Perfon to whom thofe two Chances are thus refigned has now three Chances that favour him, and confequently has now a right triple of that which he had before, and thereiore his expedation is now worth — of 100 ^• s Now if we confider that the fradion — exprefies the Probability of oDtaining the Sum of 100 ^-j and that -7- of 100, is the fame thing as -^ multiplied by 100, we muft naturally fall into this con- clufion, which has been laid down as a principle, that the value of the Expedation of any Sum, is determined by multiplying the Sum expeded by the Probability of obtaining it. This manner of reafoning, tho' deduced from a particular cafe, will eafily be perceived to be general, and applicable to any other eafe. B 2 COROL- 4 The Doctrine o/" Chances. Corollary. From what precedes, it neceflarily follows that if the Value of an Expedlation be given, as alfo the Value of the thing expedled, then dividing the firft value by the fecond, the quotient will exprefs the Probability of obtaining the Sum expefted : thus if I have an Expedlation worth 60 ^- and that the Sum which I may obtain be worth 100 '• the Probability of obtaining it will be expreft by the quotient of 60 divided by 100, that is by the fradion -^ or — • 6. The Rifk of lofing any Sum is the reverfe of Expedation ; and the true meafure of it is, the product of the Sum adventured multi- plied by the Probability of the Lofs. 7. Advantage or Difadvantage in Play, refults from the combi- nation of the feveral Expedations of the Gamefters, and of their fe- veral Rilks. Thus fuppofing that A and B play together, that A has depofited 5^ and B 3 ^- that the number of Chances which A has to win is 4, and the number of Chances which 5 has to win is 2, and that it were required in this circumflance to determine the advantage or difadvantage of the Adventurers, we may reafon in this manner : Since the whole Sum depofited is 8, and that the Probability which A has of getting it is -—- , it follows that the Expedtation of A upon the whole Sum depofited is 8 x-^= 5 -^, and for the fame reafon the Expedation of £ upon that whole Sum depofited is 8 x -|-=- 2 — . Now, if from the refpedive Expedations which the Adventurers have upon the whole fum depofited, be fubtraded the particular Sums which they depofit, that is their own Stakes, there will remain the Advantage or Difadvantage of either, according as the difference is pofitive or negative. And therefore if from 5—, which is the Expedation oi A upon the whole Sum depofited, 5 which is his own Stake, be fubtraded, there will remain — for his advantage ; likewife if from 2 — which is the Expedation of B, 3 which is his own Stake be fubtraded, there will remain — , which being negative fhews that his Dif- advantage is — . Thefe conclufions hiay alfo be derived from another confideration ; for if from the Expedation which either Adventurer has upon the Sum The Doctrine o/^ Cha NCEs. 5 Sum depofited by his Adverfary, be fubtraded the Riik of what he himfelf depofits, there will likewife remain his Advantage or Dif- advantage, according as the difference is pofitive or negative. Thus in the preceding cafe, the Stake of B being 3, and the Pro- bability which A has of winning it, being — , the Expedation of A upon that Stake is 3 x —- "= 2 ; moreover the Stake of A be- ing 5, and the Probability of lofing it, being — , his Riik ought to be eflimated by 5 x -^ ^= i — ; wherefore, if from the Expeda- tion 2, the Rifle I — be fubtraded, there will remain — as before 3 3 _ for the Advantage of A : and by the fame way of proceeding, the Difadvantage of B will be found to be — . It is very carefully to be obferved, that what is here called Advan- tage or Difadvantage, and which may properly be called G.un or Lofs, is always eftimated before the Event is come to pafs ; and altho' it be not cuftomary to call that Gain or Lof; which is to be derived from an Event not yet determined, neverthelefs in the Dodrine of Chances, that appellation is equivalent to what in common difcourfc is called Gain or Lofs. For in the fame manner as he who ventures a Guinea in an equal Game may, before the determination of the Play, be faid to be pofleflbr of that Guinea, and may, in confideration of that Sum, refign his place to another ; fo he may be faid to be a Gainer or Lofer, who would get fome Profit, or fuffer fome Lofs, if he would fell his Expedation u-::on equitable terms, and fecure his own Stake for a Sum equal to the Rifk of lofing it. 8. If the obtaining of any Sum requires the happening of feveral Events that are independent on each other, then the Value of the Expectation of that Sum is found by multiplying together the feveral Probabilities of happening, and again multiplying the produd by the Value of the Sum expcdcd. Thus fuppofing that in order to obtain 90 ^ two Events mufl happen; the firfl: whereof has 3 Chances to happen, and 2 to fail, the fecopd has 4 Chances to happen, and 5 to fail, and 1 would know the value of that Expectation ; I fay. The Probability of the firft's happening is -^ , the Probability of the fecond's happening is -— ; now multiplying thefe two Probabili- ties together, the produd will be -!i- or — ; and this produd being 45 '5 again 6 The Doctrine i^/" Chances. again multiplied by 90, the new produdt will be-^^ or 24, there- 's fore that Expedation is worth 24 ^• The Dcmonftration of this will be very eafy, if it be confider'd, that fuppofing the firft Event had happened, then that Expedation depending now intirely upon the fecond, would, before the determi- nation of the fecond, be found to be exa<flly worth — x 90 ^- or 40 ^- (by Art. 5''') We may therefore look upon the happening of the firft, as a condition of obtaining an Expedtation worth 40^- but the Probability of the firft's happening has been fuppofcd — r- , where- fore the Expectation fought for is to be eftimated by -7- x 40, or by -^ X — X 90 ; that is, by the produdl of the two Probabilities of happening multiplied by the Sum expeded. And likewife, if an Expedtation depends on the happening of one Event, and the failing of another, then its Value will be the produdl of the Probability of the firft's happening by the Probability of the fecond's failing, and of that again by the Value of the Sum ex- pedled. And again, if an Expedtation depends on the failing of two Events, the Rule will be the fame ; for that Expedtation will be found by multiplying together the two Probabilities of failing, and multiplying that again by the Value of the Sum expedted. And the fame Rule is applicable to the happening or failing of as many Events as may be affigned, COROL L AR Y. If wc make abflradtion of the Value of the Sum to be obtained, the bare Probability of obtaining it, will be the produdt of the feveral Probabilities of happening, which evidendy appears from this %^^ Art. and from the Corollary to the 5^^. Hitherto, I have confined myfelf to the confideration of Events independent ; but for fear that, in what is to be faid afterwards, the terms independent or dependent might occafion fome obfcurity, it will be necefTary, before I proceed any farther, to fettle intirely the notion of thofe terms. Two Events are independent, when they have no connexion one with the other, and that the happening of one neither forwards nor obftrudts the happening of the other. Two Events are dependent, when they are fo connedted together as * that the Probability of cither's happening is altered by the happening of tlie other. In 7^^ DoCTRINEtf/^CHANCES. 7 In order to illuftrate this, it will not be amifs to propofe the two following eafy Problems. 1°. Suppofe there is a heap of 13 Cards of one colour, and an- other heap of 13 Cards of another colour, what is the Probability that taking a Card at a venture out of each heap, I fhall take the two Aces ? The Probability of taking the Ace out of the firft heap is — : 13 now it being very plain that the taking or not taking the Ace out of the firft heap has no influence in the taking or not taking the Ace out of the fcxond ; it follows, that fuppofing that Ace taken out, the Probability of taking the Ace out of the fecond will alfo be - i and therefore, thofe two Events being independent, the Probability of their both happening will be -^ v — ^ = -^. 2"'. Suppofe that out of one fingle heap of 1 3 Cards of one colour, it fhould be undertaken to take out the Ace in the firft place, and then the Deux, and that it were required to aflign the Probability of doing it; we are to confider that altho" the Probability of the Ace's being in the firft place be ■— , and that the Probability of the Deux's being in the fecond place, would alfo be — , if that fecond Event were confidered in itfelf without any relation to the firft ; yet that the Ace being fuppofed as taken out at firft, there will remain but 1 2 Cards in the heap, and therefore that upon the fuppofition of the Ace being taken out at firft, the Probability of the Deux's being next taken will be alter'd, and become -^ -, and therefore, we may con- clude that thofe two Events are dependent, and that the Probability of their both happening will be -^ x — = ~ . From whence it may be inferred, that the Probability of the hap- pening of two Events dependent, is the produdl of the Probability of the happening of one of them, by the Probability which the other will have of happening, when the firft is confidered as having happened ; and the fame Rule will extend to the happening of as many Events as may be afligned. 9. But to determine, in the eafieft manner poffible, the Probability of the happening of feveral Events dependent, it will be convenient to diftinguifti by thought the order of thofe Events, and to fuppofe one of them to be the firft, another to be the fecond, and fo on i which being done, the Probability of the happening of the firft may be 8 iTDe Doctrine of Chances. be looked upon as independent, the Probability of the happening of the fecond, is to be determined from the fuppofition of the firfl's having happened, the Probability of the third's happening, is to be determined from the fuppofition of the firfl: and fecond having happened, and fo on: then the Probability of the happening of them all will be the product of the Multiplication of the feveral Probabili- ties which have been determined in the manner prefcribed. We had feen before how to determine the Probability of the hap- pening or failing of as many Events independent as may be afligned; we have feen likewife in the preceding Article how to determine the Probability of the happening of as many Events dependent as may be affigned : but in the cafe of Events dependent, how to determine the Probability of the liappening of fome of them, and at the fame time the Probability of the failing of fome others, is a difquifition of a greater degree of difficulty ; which for that reafon will be more conveniently transferred to another place. JO. If I have feveral Expedtations upon feveral Sums, it is very evident that my Expedtation upon the whole is the Sum of the Ex- pedations I have upon the particulars. Thus fuppofe two Events fuch, that the firft may have 3 Chances to happen and 2 to fail, and the fecond 4 Chances to happen and 5 to fail, and that I be intitled to 90 ^- in cafe the firft happens, and to an- other like Sum of 90^^ in cafe the fecond happens alfo, and that I would know the Value of my Expedtation upon the whole : I fay, The Sum expedted in the firft cafe being 90^ and the Probability of obtaining it being -7, it follows that my Expedlation on that account, is worth 90 x — r= 54 ; and again the Sum expedted in the fecond cafe being 90, and the Probability of obtaining it being — , it follows that my Expedtation of that fecond Sum is worth 90 X - = 40 ; and therefore my Expedlation upon the whole is worth 54 ^- -|- 40 ^- r= 94 ^■ But if I am to have 90^ once for all for the happening of one or the other of the two afore-mentioned Events, the method of pro- cefs in determining the value of my Expedlation will be fomewhat altered : for altho' my Expedlation of the firft Event be worth 54^ as it was in the preceding Example, yet I confider that my Expec- tation of the fecond will ceafe upon the happening of the firft, and that therefore this Expedlation takes place only in cafe the firft does happen to fail. Now the Probability of the firft's failing is - ; and fuppofing 1^)6 Doctrine c/ Chances. 9 fuppofing it has failed, then my Expeflation will be 40 ; where- fore - being the meafure of the Probability of my obtaining an Expectation worth 40 ^., it follows that this Expedation (to eftimate it before the time of the firfl's being determined) will be worth 40 X - = 16, and therefore my Expedlation upon the whole is worth 54^ -\- 1 6^= 70'-- If that which was called the fecond Event bs now confidered as the firft, and that which was called the firft be now confidered as the fecond, the conclufion will be the fame as before. In order to make the preceding Rules familiar, it will be conve- nient to apply them to the Solution of fome eafy cafes, fuch as are the following. CASE P To find the Probability of throwing an Ace in two throws of one Die. SOLUT ION. The Probability of throwing an Ace the firft time is — ; where- fore — is the firft part of the Probability required. If the Ace be miffed the firft time, ftill it may be thrown on the fecond, but the Probability of miffing it the firft time is \ , and -the Probability of throwing it the fecond time is — ; wherefore the Probability of miffing it the firft time and throwing it the fecond, is -^ X T- = —^ '• and this is the fecond part of the Probability re- quired, and therefore the Probability required is in all . — -| \- ■=. 1 1 To this aife is analogous a queftion commonly propofed about throwing with two Dice either fix or feven in two throws ; which will be eafily folved, provided it be known that Seven has 6 Chances to come up, and Six 5 Chances, and that the whole number of Chances in two Dice is 36 : for the number of Chances for throw- ing fix or feven being 1 1 , it follows tlwt the Probability of throwing either Chance the firft time is -— : but if both are miffed the firft time, ftill either may be thrown the fecond time 3 now the Proba- C bility 10 The Doctrine <?/ Chances. bility of miffing both the firft time is -^ , and the Probabili- ty of throwing either of them the fecond time is ~ : wherefore the Probability of miffing both of them the firfl: time, and throw- 2C II 271; ing either of them the fecond time, is — ■ x — = — ^ , and therefore the Probability required is — -J- -^^ -.= — '— ^ and the Probabihty of the contrary is — ^ . •' ^ 1290 CASE IP 7oJi?id the Probability of throwing an Ace in three throws. Solution. The Probability of throwing an Ace the firfl time is — , which is the firft part of the Probability required. If the Ace be miffed the firft time, ftill it may be thrown in the two remaining throws ; but the Probability of miffing it the firfl -time is \ , -and the Probability of throwing it in the two remain- ing times is (by Cafe i^) = -— ■ . And therefore the Probability of miffing it the firft time, and throwing it in the two remaining times '^ b" 5^ "~6" ^^^ Tif » "which is the fecond part of the Probability required j wherefore the Probability required will be 5 -^ ^ = CASE IIP- "To find the Probability of throwing an Ace in four throws. Sol ution. The Probability of throwing an Ace the firft time is - , which is the firft part of the Probability required. If the Ace be miffed the firft time, of which the Probability is \ , there remains the Probability of throwing it in three times, which (by Cafe 2^) is -^ ; wherefore the Probability of miffing the Ace the firft time, and throwing it in the three remaining times, \i&=r -r X -^ = -^ , which is the fecond part of the Proba- ^ bility 7^^ Doctrine <?/'Chances., fx billty required ; and therefore the Probability required is, in the whole, -i- + 1^ = -^ , and the Probability of the contrary -^. It is remarkable, that he who undertakes to throw an Ace in four throws, has juft the fame Advantage of his adverfary, as he who undertakes with two Dice that fix or feven (hall come up in two- throws, the odds in either cafe being 671 to 625: whereupon it will not be amifs to (hew how to determine eafily the Gain of one Party from the Superiority of Chances he has over his adverfary, upon fuppofition that each flake is equal, and denominated by unity. For although this is a particular cafe of what has been ex- plained in the 7''' Article ; yet as it is convenient to have the Rule ready at hand, and that it be eafily remembered, I (Ivall fet it down here. Let therefore the odds be univerfally expreflfed by the ratio of a to l>, then the refpedlive Probabilities of winning being - b b and — — J- , the right of the firft upon the Stake of the fecond is — ^— X I, and like wife the right of the fecond upon the Stake of the firfl: is _.L^ x i, and therefore the Gain of the firft is °~, X i a-\-b ' a + b or barely -~ , : and confequently the Gain of hin\ who under- takes that fix or feven (hall come up in two throws, or who under- . 671 — 6z; 46 , takes to ning aa Ace in, four throws, is ' 071+625" ^^ TI^ » ^"^' is nearly -^ part of his adverfary 's Stake. CASE IV'^. 'To find the Probability of throwing two Aces in two throws. Solution. It is plain (by the S'"" Art.) that the Probability required is - X - = -i- . 6 (J 3O ' CASE V^. To find the Probability of throwing two Aces ift three throws. Solution. If an Ace be thrown the firft time, then it will only be required to throw it once in two throws ; but the Probability of throwing it the firft time is ^ , and the Probability of throwing it once in C 2 two 12 The Doctrine 0/ Chances. two throws (by the firft cafe) is -r^ : wherefore the Probability of throwing it the firft time, and then throwing it once in the two remaining times is ^ X ^5"= 77^ ; and this is equal to the firfl part of the Probability required. If the Ace be miffed the firft time, ftill there remains the Proba- bility of throwing it twice together, but the Probability of miffing it the firft time is -^ , and the Probability of throwing it twice together is (by the 4*'^ Cafe) = -1- ; therefore the Probability of 3'- 5 both Events is L '^i^ 1. z=—. ; which is the fecond part of 6 36 216 ^ the Probability required: therefore the whole Probability required. ^ 216 ^^^ Tib ■ CASE vr\ To find theProbability of throwing two Aces in four throws. Solution. If an Ace be thrown the firft time, no more will be required than throwing it again in three throws ; but the Probability of throwing an Ace the firft time is -^ , and the Probability of throw- ing it in three times is ^ (by the 2^ Cafe j) wherefore the Proba- bility of both happening is -^ x -^ = -—■ =1" part of the Probability required. If the Ace be miffed the firft time, ftill there will remain the- Probability of throwing two Aces in three throws ; but the Proba- bility of miffing the Ace the firft time is — , and the Probability 111 of throwing it twice in three throws is -^ , (by the f'*" Cafe;) wherefore the Probability of both together is -i x -j;-; = 77—- = 2^ part of the Probability required: and therefore the Probability required = -^ -j- — = -^ . And, by the fame way of reafoning, we may gradually find the Probability of throwing an Ace as many times as fhall be demanded,. in any given number of throws. If, inftead of employing figures in the Solutions of the foregoing Cafes, we employ algebraic Characters, we. fliall readily perceive a. moft regular order in thofe Solutions. !'•- 7^^ Doctrine (?/" Chances. 13 r I. Let therefore a be the number of Chances for the happening of an Event, and b the number of Chances for its failing ; then thS Probability of its happening once in any number of Trials will b*" exprefled by the Series " , ab , gib ai! T^' ^ T77\= + TTTi' + a^b]"' ' ^^- "^^^^^ ^^'■^^s is to be continued to fo many terms as are equal to the number of Trials given : thus if a be= I, b = s, and the number of Trials given =4, then the Probability required will be exprefled by -^ -}- -^ _|- -ii- 1 '- f 71 ' 1:96 The fame things being fuppofed as before, the Probability of the Event's happening twice in any given number of Trials, will be ex- prefled by the Series ^\ 4. '^"^^ 4. '"''^'' 1 ^f^fJ , c.«M . ,. 7^\ ^^ , 7^* ^ ~^^^ '^ ==-rv , ccc. Which is to be continued to fo many term?, wanting one, as is the number of Trials given ; thus let us fuppofe a = 1 S = s, and the number of Trials 8, then the Probability required will he expreflid by 7— 4- -^ J- -^^ 4. -i£2_ 1 ^ '- _, 10750 . ioq"i; nb^QQt 279936 ~' 1679616 1079616 And again, the Probability of the Event's happening three times in any given number of Trials will be exprefled by the Series y+K^'^'TTlY'^TTlV"^ TTW^T^A^' ^^- which is to be continued to fo many terms, wanting two, as is the num- ber of terms given. But all thefe particular Series may be comprehended under a ge- neral one, which is as follows. Let a be the number of Chances, whereby an Event may hap- pen, b the number of Chances whereby it may fail, / the num- ber of times that the Event is required to be produced in any given number of Trials, and let « be the number of thofe Trials j make a -^ b = .c, then the Probability of the Event's happen- ing / times in « Trial?, will be cxprefl'ed by the Series —j- x "l~ / ~l~ 1. 2. SJ I 1. 2. 3. j! I 1. i. 3. 4 1* &c which Series is to be continued to fo many terms exclufive of thc- It is to be noted here, and elfewhere, that the points here made vfe of, Jiand tnjlead of the Mark of Multiplication X. 14 72^ Doctrine of Chaftces^ the common multiplicator — as are denoted by the number n — And for the fame reafon, the Probability of the contrary, that is of the Event's not happening fo often as / times, nia- ,p king n — / -j- i r=. p, will be exprefled by the Series — x jP , J>a_ , p.p-^-i- aa i P P+ ' ■ /■ + 2. "' i t- P + ■•/'4-Z-/'+ 3- "* "1" J ~r ,. 2.. JJ "T~ I. 2. 3. j' ~l" I. 2. 3, 4. j4 > vs'hich Series is to be continued to fo many terms, exclufive of the common multiplicator, as are denoted by the number /. Now the Probability of an Event's not happening being known, the Probability of its happening will likewife be known, fince the Sum of thofe two Probabilities is always equal to Unity ; and there- fore the fecond Series, as well as the firft, may be employed in determining the Probability of an Event's happening : but as the number of terms to be taken in the firft is expreffed by n — / -\- i, and the number of terms to be taken in the fecond is expreffed by /, it will be convenient to ufe the firft Series, if n — / -]- i be lefs than /, and to ufe the fecond, if / be lefs than « — I -\- i; ot in other terms, to ufe the firft or fecond according as / is lefs or greater than — ^ — . Thus, fuppofc an Event has i Chance to happen, and 3 5 to fail, and that it were required to affign the Probability of its happening in 24 Trials; then becaufe in this cafe h = 24 and / = i, it is plain that 24 terms of the firft Series would be requifite to anfwer the Queftion, and that one fingle one of the fecond will be fuffi- cient : and therefore, if in the fecond Series we make ^^35, tf = I, and /= 1, the Probability of the Event's not happening once in 24 Trials, will be expreffed by —^ x i, which by the help of Logarithms, we ftiall find nearly equivalent to the decimal fradlion 0.50871 ; now this being fubtraded from Unity, the re- mainder 049129 will exprefs the Probability required ; and there- fore the odds againft the happening of the Event will be 50 to 49 nearly. Again, fuppofe it be required to aflign the Probability of the pre- ceding Event's happening twice in 60 Trials ; then becaufe / = 2, and ;; = 60, n — / + ^ will be = ^g, which fhews that 59 terms of the firft Series would be required : but if we ufe the fecond, then by reafon of / being = 2, two of its terms will be fufficient ; wherefore 1*1:16 Doctrine (7/*Chances. 15 wherefore the two terms -—^ X i + — ^ will denote the Proba- bility of the Event's not happening twice in 60 Trials. Now re- ducing this to a decimal fradion, it will be found equal to 0.5007, which being fubtradlcd from Unity, the remainder 0.4993 will ex- prefs the Probability required ; and therefore the odds againft the Event's happening twice in 60 times will be very little more than 500 to 499. It is to be obferved of thofe Series, that they are both derived from the fame principle ; for fuppofing two adverfaries A and 5, contending about the happening of that Event which has every time a chances to happen, and b chances to fail, that the Chances a are favourable to A^ and the Chances b to B, and that A fhould lay a -wager with 5, that his Chances fhall come up / times in n Trials : then by reafon B lays a wager to the contrary, he himfelf under- takes that his own Chances fliall, in the fame number of Trials, come up ?? — I ■\- I times j and therefore, if in the firfl: Series, wc change / into ;z — I -\- j, and vice 'versa, and alfo write b for a, and a for b, the fecond Scries will be formed. It will be eafy to conceive how it comes to pafs, that if A un- dertakes to win / times in n Trials, his Adverfary B neceflarily un- ,dertakes in the fame number of Trials to win ?i — I -\- 1 times, if it be confidered that A lofes his wager if he wins but / — i times ; now if he wins but / — i times, then fubtrafting / — i from //, the remainder fhews the number of times that B is to win, which therefore will be « — / -\- 1. CASE vir'' 2o ji?id the Probability of' throwing one Ace^ and no morey ill four throws. Solution. This Cafe ought carefully to be diftinguiflied from the fourth 1 for there it was barely demanded, without any manner of reftridtiop, what the Probability was of throwing an Ace in 4 throws j now in- this prefent cafe there is a reflraint laid on that Event : for whereas in the former cafe, he who undertakes to throw an Ace defifts from throwing when once the Ace is come up ; in this he obliges himfelf, after it is come up, to a farther Trial which is wholly againft him, excepting the laft throw of the four, after which there is no Trial j and therefore we ought from the unlimited Probability of the 1 6 The Doctrine <7/'Chances. the Ace's being thrown once in 4 throws, to fubtradl the Probability of its being thrown twice in that number of throws : now the firft Probability is -7^ (by the 3^^ Cafe, and the fecond Probability is (by the G-^ Cafe,) from which it follows that the Probability re- '"'-'^ -00 796 quired is -—- , and the Probability of the contrary 7—^; and therefore the Odds againft throwing one Ace and no more in 4 throws are 796 to 500, or 8 to 5 nearly : and the fame method may be follow'd in higher cafes. CASE VIII'' If A and B play together^ and that A wants but i Game of being up^ and B wants 2 ; what are their refpe&ive Probabilities of wiftning the Set f Solution. It is to be confidered that the Set will necellarily be ended in two Games at moft, for if A wins the firft Game, there is no need of any farther Trial ; but if B wins it, then they will want each but 1 Game of being up, and therefore the Set will be determined by the fecond Game : from whence it is plain that A wants only to win once in two Games, but that B wants to win twice together. Now fuppofing that A and B have an equal Chance to win a Game, then the Probability which B has of winning the firft Game will be — , and confequently the Probability of his winning twice toge- ther will be — X — =: — 5 and therefore the Probability which 224 A has of winning once in two Games will be i = — , from D 44 whence it follows that the Odds oi A^ winning are 3 to i. CASE IX'''- A and B play together^ A wa?tts i Game of being up^ and B wants 2 ; but the Chances whereby B may win a Game^ are double to the nu?7iber oj Chajices whereby A 7nay wi?i the fame : 'tis required to ajftgn the refpeBive Probabi- lities of wi7i7iing. Solution. It is plain that in this, as well as in the preceding cafe, B ought to win twice together j now fince B has 2 Chances to win a Game, and The Doctrine <?/'Chances. 17 and A i Chance only for the fame, the Probability which B has of winning a Game is — , and therefore the Probability of his winning twice together is — x "~ = " > and confequently the Probability of A' winnins; the Set is i = — ; from whence it follows that the Odds of ^^ winning once, before B twice, are as 5 to 4. Remark. Altho' the determining the precife Odds in queftions of Chance requires calculation, yet fometimes by a fuperficial View of the quef- tion, it may be poffible to find that there will be an inequality in the Play. Thus in the preceding cafe wherein B has in every Game twice the number of Chances of A^ if it be demanded whether A and B play upon the fquare, it is natural to confider that he who has a double number of Chances will at long run win twice as often as his Adverfary ; but that the cafe is here otherwife, for B under- taking to win twice before A once, he thereby undertakes to win oftner than according to his proportion of Chances, (ince A has a right to expedl to win once, and therefore it may be concluded that B has the difadvantage : however, this way of arguing in gene- ral ought to be ufed with the utmofl: caution. 12. Whatever be the number of Games which A and B refpec- tively want of being up, the Set will be concluded at the moft in fo many Games wanting one, as is the fum of the Games wanted between them. Thus fuppofe that A wants 3 Games of being up, and 5 5 ; it is plain that the greateft number of Games that A can win of B before the determination of the Play will be 2, and that the greateft number which B can win of A before the determination of the Play will be 4 ; and therefore the greateft number of Games that can be played between them before the determination of the Play will be 6 : tut fuppofing they have played fix Games, the next Game will terminate the Play ; and therefore the utmoft number of Games that can be played between them will be 7, that is one Game lefs than the Sum of the Games wanted between them. D CASE 1 8 iTje Doctrine 0/ Chances. CASE X"^ Suppojtng that A wajiti 3 Games of being tip, and 'Q wants 7 ; but that the Chafices which A a?id B re/pe&ively have for winning a Gaitie are as t, to ^, to find the re- JpeSiive Probabilities of winning the Set. Solution. By reafon that the Sum of the Games wanted between A and B is 10, it is plain by the preceding Paragraph that the Set will be con- cluded in 9 Games at moft, and that therefore A undertakes out of 9 Games to win 3, and B, out of the fame number, to win 7; now fuppofing that the firft general Theorem laid down in the 1 1 '*" Art. is particularly adapted to reprefent the Probability of A"^ win- ning, then / = 3 ; and becaufe n reprefents the number of Games in which the Set will be concluded, ?z =r 9 ; but the number of terms to be ufed in the firfl Theorem being = « — I -\- 1 = 7, and the number of terms to be ufed in the fecond Theorem being = /= 3, it will be more convenient to ufe the fecond, which will re- prefent the Probability of B^ winning. Now that fecond Theorem being applied to the cafe of n being = 9, /= 3, <z = 3, b =5, the Probability which 5 has of winning the Set will be exprefled by 1^x14-^ + ~ =-g-X484= 0.28172 nearly; and therefore fubtradting this from Unity, there will remain the Proba- bility which yf has of winning the fame, which will be :=z 0.71828 : and confequently the Odds of A^ winning the Set will be 71828 to 28172, or very near as 23 to 9. The fame Principles explained in a different and more general iv ay. Altho' the principles hitherto explained are a fufficient introdudtion to what is to be faid afterwards ; yet it will not be improper to rc- fumc fome of the preceding Articles, and to fet them in a new light : it frequently happening that fome truths, when reprefented to the mind under a particular Idea, may be more eafily apprehended than when reprefented under another. 13. Let us therefore imagine a Die of a given number of equal faces, let us likewife imagine another Die of the fame or any other numbtr of equal faces ; this being fuppofed, I fay that the number of all the variations which the two Dice can undergo will be obtained by multiplying the number effaces of the one, by the number of faces of the other. In Ihe Doctrine (?/"Chances. 19 In order to prove this, and the better to fix the imagination, let us take a particular cafe : Suppofe therefore that the firft Die con- tains 8 faces, and the fecond \z -, then fuppofing the firft Die to ftand ftill upon one of its faces, it is plain that in the mean time the fecond Die may revolve upon its 1 2 faces ; for which reafon, there will be upon that fingle fcore 1 2 variations : let us now fup- pofe that the firft Die ftands upon another of its faces, then v.hiiil that Die ftanJs ftill, the fecond Die may revolve again upon its 12 faces ; and fo on, till the faces of the firft Die have undergone all their changes : from whence it follows, that in the two Dice, there will be as many times 1 2 Chances as there are faces in the firft Die ; but the number of faces in the firft Die has been fuppofed 8, where- fore the number of Variations or Chances of the two Dice will be 8 times 12, that is 96: and therefore it may he univerfally con- cluded, that the number of all the variations of two Dice will be the produdt of the multiplication of the number of faces of one Die, by the number of faces of the other. 14. Let us now imagine that the faces of each Die are diftin- guiftied into white and black, that the number of white faces upon the firft is A, and the number of black faces B, and alfo that the number of white faces upon the fecond is a, and the number of black faces b ; hence it will follow by the preceding Article, that multiplying A -I- B by <z -}- b^ the produdt ha -\- hb -\- B<z -\- '^b, will exhibit all the Variations of the two Dice : Now let us fee what each of thefe four parts feparately taken will reprefent. 1°. It is plain, that in the fame manner as the produd of the multiplication of the whole number of faces of the firft Die, by the whole number of faces of the fecond, exprefles all the variations of the two Dice ; fo likewife the multiplication of the number of the white faces of the firft Die, by the number of the white faces of the fecond, will exprefs the number of variations whereby the two Dice may exhibit two white faces : and therefore, that number of Chances will be reprefented by h.a. 2°. For the fame reafon, the multiplication of the number of white faces upon the firft Die, by the number of black faces upon the fecond, will reprefent the number of all the Chances whereby a white face of the firft may be joined with a black face of the fecond ; which number of Chances will therefore be reprefented by Ab. ^ 3°. The multiplication of the number of white faces upon the fecond, by the number of black faces upon the firft, will exprefs the number of all the Chances whereby a white face of thj fecond D 2 may 20 Ihe Doctrine o/'Chances. may be joined with a black face of the firft j which number of Chances will therefore be reprefentecl by aB. 4°. The multiplication of the number of black faces upon the firft, by the number of black faces upon the fccond, will exprefs the number of all the Chances whereby a black face of the firft may be joined with a black face of the fecond ; which number of Chances will therefore be reprefented by B^. And therefore we have explained the proper fignification and ufe of the feveral parts ha, hb, Ba, B^^ fingly taken. But as thefe parts may be connefted together feveral ways, fo the Sum of two or more of any of them will anfwer fome queftion of Chance : for inftance, fuppofe it be demanded, what is the number of Chances, with the two Dice above-mentioned, for throwing a white face ? it is plain that the three parts Ka -|- Ab -\- Ba will anfwer the queftion ; iince every one of thofe parts comprehends a cafe wherein a white face is concerned. It may perhaps be thought that the firft term A^ is fuperfluous, it denoting the number of Variations whereby two white faces can be thrown ; but it will be eafy to be fatisfied of the neceflity of taking it in : for fuppofing a wager depending on the throwing of a white face, he who' throws for it, is reputed a winner, whenever a white face appears, whether one alone, or two together, unlefs it be exprefly ftipulated that in cafe he throws two, he is to lofe his wager; in which latter cafe the two terms Kb -|- B<z would repre- fent all his Chances. If now we imagine a third Die having upon it a certain number of white faces reprefented by «, and likewife a certain number of black faces reprefented by /3, then multiplying the whole variation of Chances of the two preceding Dice viz. ha + hb -\-'Ba -\- B/^ by the whole number of faces a -|- /3 of the third Die, the produdl had. + hbcc -1- Baa + Bbot + Art/3 -1- A/;/3 + Brt/3 \ BbjS will ex- hibit the number of all the Variations which the three Dice can un- dergo. Examining the feveral parts of this new produft, we may eafily perceive that the firft term hax repiefents the number of Chances for throwing three white faces, that the fecond term Ab» repre- fents the number of Chances whereby both the firft and third Die may exhibit a white fecc, and the fecond Die a black one ; and that the reft of the terms have each their particular properties, which are difcovered by bare infpedlion. ^ It may alfo be perceived, that by joining together two or more of thofe terms, fome queftion of Chance will thereby be anfwered : for inftance, Ithe Doctrines/ Chances. 2r inftance, if it be demanded what is the number of Chances for throwing two white faces and a black one ? it is plain that the three terms kbu, B^a, A^/S taken together will exhibit the number of Chances required, fince in every one of them there is the expreffion of two white faces and a black one ; and therefore if there be a wager depending on the throwing two white faces and a black one, he who undertakes that two white faces and a black one fliall come up, has for him the Odds of kba. + V>aa. -j- Atf/3 to hau + '^boc + hb^ -\- Btf/3 \ Bbfi ; that is, of the three terms that include the con- dition of the wager, to the five terms that include it not. When the number of Chances that was required has been found, then making that number the Numerator of a fradlion, whereof the Denominator is the whole number of variations which all the Dice can undergo, that fradtion will exprefs the Probability of the Event j as has been fliewn in the firft Article. Thus if it was demanded what the Probability is, of throwing three white faces with the three Dice above-mentioned, that Probability will be exprelTed by the fradion ^ ^_^ ^ . ^ b ..+ aJ^uu + A.g-f B.^g+ ^"0 ; But it is to be obferved, that in writing the Denominator, it will be convenient to exprefs it by way of producft, diftinguifliing the feveral multiplicators whereof it is compounded ; thus in the preced- ing cafe the Probability required will be befl exprelTed as follows. A /^cc If the preceding fraction be conceived as the produdl of the three fradions ■- , , x — ^^ X — ^^r , whereof the firft exprefles the Probability of throwing a white face with the firft Die ; the fecond the Probability of throwing a white face with the fecond Die, and the third the Probability of throwing a white face with the third Die ; then will again appear the truth of what has been demonftra- ted in the 8"* Art. and its Corollary, viz. that the Probability of the happening of feveral Events independent, is the produdl of all the particular Probabilities whereby each particular Event may bs produced ; for altho' the cafe here defcribed be confined to three Events, it is plain that the Rule extends itfelf to any number of them. Let us refume the cafe of two Dice, wherein we did fuppofe that the number of white faces upon one Die was expreffed by A, and the number of black faces by B, and alfo that the number of white faces upon the other was exprefTed by a, and the number of black faces by b, which gave us all the variations Aa -\- Ab -\- aB -\- Bb i and 2 2 T^he Doctrine o/" Chances. and let us imagine that the number of the white and black, faces is re- fpeftively the fame upon both Dice : wherefore k-z^a, and B =r />, and confequently inftead of h.a ■\- hb -\- <jB -f- B/^, we fliali have aa -\- ab -\- ab -j- bb^ or aa -f- ^ab -)- bb ; but in the firft; cafe hb -\- rtB did exprefs the number of variations whereby a white face and a black one might be thrown, and therefore lab which is now fub- ftituted in the room of hb -\- «B does exprefs the number of varia- tions, whereby with two Dice of the fame refpeftive number of white and black faces, a white face and a black one may be thrown. In the fame manner, if we refume the general cafe of three Dice, and examine the number of variations whereby two white faces and a black one may be thrown, it will eafily be perceived that if the number of white and black faces upon each Die are refpedively the fame, then the three parts kba. -\- Bacx. -\- Arf/3 will be changed into aba -{- baa -\- ^<^b. or "T^aab, and that therefore ^aab, which is one term of the Binomial a -\- b raifed to its Cube, will exprefs the number of variations whereby three Dice of the fame kind would exhibit two white faces and a black one. 1 5. From the preceding confiderations, this general Rule may be laid down, viz. that if there be any number of Dice of the fame kind, all diftinguifhed into white and black faces, that 71 be the num- ber of thofe Dice, a and b the refpedtive numbers of white and black faces upon each Die, and that the Binomial ^ -f- ^ be raifed to the power n ; then 1°, the firft term of that power will exprefs the number of Chances whereby « white faces may be thrown ; 2^, that the fecond term will exprefs the number of Chances whereby n — i white feces and i black face may be thrown ; 3 ', that the third term will exprefs the number of Chances whereby n — 2 white faces and 2 black ones may be thrown ; and fo on for the reft of the terms. Thus, for inftance, if the Binomial a -\- b be raifed to its 6*'' power, which is a^ -\- ba^b -\- i^a'^b'- -{- 2oa}b^ -\- i^a'^b'^ -|- bab"^ -f b^ ; the firft term a^ will exprefs the number of Chances for throwing 6 white faces ; the fecond term ba'^b will exprefs the num- ber of Chances for throwing 5 white and i black ; the third term I z,a^b'^' will exprefs the number of Chances for throwing 4 white and 2 black; the fourth iQa'^b'^ will exprefs the number of Chances for throwing 3 white and 3 black; the fifth ij^^M will exprefs the number of Chances for throwing two white and 4 black ; the fixth tab'' will exprefs the number of Chances for 2 white and 4 black ; laftly, the feventh b^ will exprefs the number of Chances for 6 black. And The Doctrine <?y Chances. 23 And therefore having raifed the Binomial a -\;- b X.o any given power, we may by bare infpedion determine the property of any one term belonging to that power, by only obferving the Indices wherewith the quantities a and b are affefted in that term, fince the refpedlive numbers of white and black faces are reprefented by thofe Indices. The better to compare the confequences that may be derived from the confideration of the Binomial a -\- b raifed to a power given, with the method of Solution that hath been explained before ; let us refume fome of the preceding queftions, and fee how the Binomial can be applied to them. Suppofe therefore that the Probability of throwing an Ace in four throws Vi'ith a common Die of fix faces be demanded. In order to anfwer this, it mufi: be confidered that the throwing of one Die four times fuccefTively, is the fame thing as throwing four Dice at once ; for whether the fame Die is ufed four times fuc- cefTively, or whether a different Die is ufed in each throw, the Chance remains the fame ; and whether there is a long or a fhort interval be- tween the throwing of each of thefe four different Dice, the Chance remains flili the fame ; and therefore if four Dice are thrown at once, the Chance of throwing an Ace will be the fame as that of throwing it with one and the fame Die in four fuccefiive throws. This being premifed, we may transfer the notion that was introduced concerning white and black faces, in the Dice, to the throwing or miffing of any point or points upon thofe Dice ; and therefore in the prefent cafe of throwing an Ace with four Dice, we may fuppofe that the Ace in each Die anfwer to one white face, and the reft of the points to five black faces, that is, we may fup- pofe that a r=r- 1, and ^ ;= 5 ; and therefore, having raifed a A;- b to its fourth power, which is rt"* -\- \a'^b -f- 6^'/^- -J- d^ab"- -\- b^, every one of the terms wherein a is perceived will be a part of the number of Chances whereby an Ace may be thrown. Now there are four of thofe parts into which a enters, viz. a* -\- /^a'b -\- ba^b^ •\- i^ab'^^ and therefore having made a z=i\, and ^ = 5, we fliall have I Ar 20 -\- 150 -j- 500 = 671 to exprefs the number of Chances whereby an Ace may be thrown with four Dice, or an Ace thrown in four fuccefTive throws of one fingle Die : but the number of all the Chances is the fourth power of a -\- by that is the fourth power of 6, which is 1296 ; and therefore the Probabihty required is mea- fured by the fraction -j^ , which is conformable to the refolution. given in the 3"^ cafe of the queflions belonging to the lo*** Art, It 2ij. Ihe Doctrine ^j/'Chances. It Is to be obferved, that the Solution would have been fliorter, if inftead of inquiring at firfl: into the Probability of throwing an Ace in four throws, the Probability of its contrary, that is the Probabi- lity of miffing the Ace four times fucceffively, had been inquired into : for fince this cafe is exadly the fame as that of miffing all the Aces with four Dice, and that the laft term M of the Binomial a -^-b raifed to its fourth power exprefles the number of Chances whereby the Ace may fliil in every one of the Dice j it follows, that the Probability of that failinp; is =r=rv = "TT^T" > and therefore the Probability of not failing, that is of throwing an Ace in four , . (izc 1206 — 62c 6-71 throws, is I — r= — - — r— ^ = • ' 1 zqo izg6 1290 From hence it follows, that let the number of Dice b e what it will, fuppofe ;;, then the laft term of the power a -\- b\ \ that is b\ will always reprefent the number of Chances whereby the Ace may fail n times, whether the throws be confidered as fucceffive or cotemporary : Wherefore ' ^^i;^;, is the Probability of that failing ; and confequently the Probability of throwing an Ace in a number of throws expreffed by n, will be i — ^ ■ = — , .. . Again, fuppofe it be required to affign the Probability of throw- ing with one fingle Die two Aces in four throws, or of throwing at once two Aces with four Dice : the queftion will be anfwered by help of the Binomial a -\- b raifed to its fourth power, which being a^ -^ ^a-b -|- 6rt*^* -\- ^ab^ -\- b'^, the three terms «* -|- ^a'b + ba^'b'- wherein the Indices of ^ equal or exceed the number of times that the Ace is to be thrown, ^vill denote the number of Chances where- by two Aces may be thrown ; wherefore having interpreted ^ by i, and b by 5, the three terms above-written will become i -\- 20 -\- 1 50 — 17 1, but the whole number of Chances, 'viz. a -\- b\ '^ is in this cafe — 1296. and therefore the Probability of throwing two Aces in four throws will be meafured by the fradlion -— ^ . But if we chufe to take at firft the Probability of the contrary, it is plain that out of the five terms that the fourth power of a -\- b confifts of, the two terms 4^/75 -\- b'^ ; m the lirft of which a enters but once, and in the fecond of which it enters not, will exprefs the num- ber of Chances that are contrary to the throwing of two Aces ; which pumber of Chances will be found equal to 500 -1- 625 = 1125. And therefore the Probability of not throwing two Aces in four throws The Doctrine <?/" Chances. 25 1 1 21; throws will be 7^: from whence may be deduced the Proba- bility of doing it, which therefore will be i — 1 \ 2C 1296 1 I 2? 1290 1296 = -^^ as it was found in the preceding paragraph ; and this agrees with the Solution of the fixth Cafe to be feen in the 10"' Article. Univerfally, the laft term of any power a^h^' being b\ and the laft but one being nab"—^, in neither of which a^ enters, it fol- lows that the two lafl terms of that power exprefs the number of Chances that are contrary to the throwing of two Aces, in any num- ber of throws denominated by n ; and that the Probability of throw- . .., , tiab ■+■* '? -h 01 — nab — b m? two Aces will be i — , ^>,„ = .v, . And likewife, in the three laft terms of the power a-\- B^ \ every one of the Indices of a will be lefs than 3, and confequently thofe three laft terms will ftiew the number of Chances that are contrary to the throwing of an Ace three times in any number of Trials de- nominated by n : and the fame Rule will hold perpetually. And thefe conclufions are in the fame manner applicable to the happening or failing of any other fort of Event in any number of times, the Chances for happening and failing in any particular Trial being refpeflively reprefented by a and b. 16. Wherefore we may lay down this general Maxim ; that fup- poling two Adverfaries A and B contending about the happening of an Event, whereof^ lays a wager that the Event will happen /times in « Trials, and B lays to the contrary, and that the number of Chances whereby the Event may happen in any one Trial are a, and the number of Chances whereby it may fail are^, then fo many of the laft terms of the power a -^ A" expanded, as are reprefented by /, will fhew the number of Chances whereby B may win his wager. Again, B laying a wager that A will not win / times, does the fame thing in effed as if he laid that A will not win above / — i times ; but the number of winnings and lofings between A and B is n by hypotheiis, they having been fuppofed to play n times, and therefore fubtrafting / — i from «, the remainder n — / + i will fliew that B himfelf undertakes to win n — I •{- i times ; let this re- mainder be called />, then it will be evident that in the fame manner as the laft terms of the power a -j^V expanded, viz, b" -j- E nab 26 The Doctrine ©/"Chances. nab"-^ 4- Y X-^^-r^ a^b'-"-, &c. the number whereof is /, do exprcfs the number of Chances whereby B may be a winner, fo the firft terms a" -Y na'—^b + 7 X "-^^ a —''b'', &c. the number. whereof is />, do exprefs the number of Chances whereby A may be a winner. 17. If A and B being at play, refpedively want a certain number of Games / and p of being up, and that the refpeftive Chances they have for winning any one particular Game be in the proportion of aio b', then raifing the Binomial ^ -|- ^ to a power whofe Index fhall hQ I -\- p — I, the number of Chances whereby they may refpedlively win the Set, will be in the fame proportion as the Sum of fo many of the firft terms as are exprefled by p, to the Sum of fo many of the laft terms as are exprefled by /. This will eafily be perceived to follow from what was faid in the preceding Article ; for when A and B refpedtively undertook to win / Games and p Games, we have proved that if n was the number of Games to be played between them, then p was neccflarily equal to n — / + I, and therefore I -\- pz=n -\- i, and ti=^ I -\-p — i ; and confequently the power to which a -\- b h to be raifed will be l^p—x. Thus fuppofing that A wants 3 Games of being up, and B 7, that their proportion of Chances for winning any one Game are re- fpeftively as 3 to 5, and that it were required to affign the proportion of Chances whereby they may win the Set ; then making 7=3, ^ = 7, <zr=:3, b z=i ^, and raifing a -{- b to the power denoted by I -]- p — I, that is in this cafe to the 9'*" power, the Sum of the firft I feven terms will be to the Sum of the three laft, in the proportion of the rcfpedive Chances whereby they may win the Set. Now it will be fufficient in this cafe to take the Sum of the three laft terms; for fince that Sum exprefles the number of Chances whereby B m.ay win the Set, then it being divided by the 9'^ power of a -\- b^ the quotient will exhibit the Probability of his winning ; and this Probability being fubtradled from Unity, the remainder will exprefs the Probability of A^ winning : but the three laft terms of the Binomial a -[- b raifed to its 9*'' power are b^ -|- gab^ -\- ^6aab'', which being converted into numbers make the Sum 378 12500, and the 9'^ power of a -J[- b is 1342 17728, and therefore the Probability of B'5 winning will be exprefled by the fradion "ij^^'AyTs ^^^ ■^^''^'^^- : let this be fubtradled from Unity, then the remainder •J3SS443* 24101 ^,07 33554432 7he Doctrine ©/"Chances. 27 ''^'°'^°'^ • will exprefs the Probability of A^ winnln? : and therefore the Odds of A" being up before B, are in the proportion of 24101307 to 9453 125, or very near as 23 to 9 : which agrees with the Solu- tion of the 10 "^ Cafe included in the lo'*" Article. In order to compleat the comparifon between the two Methods of Solution which have been hitherto explained, it will not be im- proper to propcfe one cafe more. Suppofe therefore it be required to aflign the Probability of throw- ing one Ace and no more, with four Dice thrown at once. It is vifible that if from the. number of Chances whereby one Ace or more may be thrown, be fubtradted the number of Chances whereby two Aces or more may be thrown, there will remain the number of Chances for throwing one Ace and no more ; and there- fore having raifed the Binomial a -1- b to its fourth power, which is «+ -|- i^a^b -|- ta-b^ -\- /^ab'^ -\- b*, it will plainly be feen that the four firft terms exprefs the number of Chances for throwing one Ace or more, and that the three firfl terms exprefs the number of Chances for throwing two Aces or more ; from whence it follows that the fingle term 4^^' does alone exprefs the number of Chances for throw- ing one Ace and no more, and therefore the Probability required will be : p TT- r= — - =r — ^ : which agrees witli the Solution of the 7"' Cafe given in the 10"* Article. This Conclufion might alfo have been obtained another way : for applying what has been faid in general concerning the property of any one term of the Binomial a -\- b raifed to a power given, it will thereby appear that the term ^ab^ wherein the indices of a and b are refpediively i and 3, will denote the number of Chances where- by of two contending parties A and B, the firfl may win once, and the other three times. Now A who undertakes that he fhall win once and no more, does properly undertake that his own Chance fhall come up once, and his adverHiry's three times; and therefore the term 4^^' expreffes the number of Chances for throwing one Ace and no more. In the like manner, if it be required to affign the Chances for throwing a certain number of Aces, and it be farther required that there fl ull no t be above that number, then one fingle term of the power ^ -\- b^ " will always anfwcr tlie queftion. Bit to find that term as expeditioufly as poffiblc, fuppofe n to be the number of Dice, and / the precife number of Aces to be thrown ; then if / be lefs than — «, write as many terms of the E 2 Series 7he Doctrine <?/"Chances. &c. as there are Units »— 3 •—4 28 Series^, -^, J , in/; or if / be greater than - n, write as many of them as there arc Units in -^ « — /j then let all thofe terms be multiplied together, and the produd be again multiplied by a'b—^ ; and this lafl: produdt will exhibit the term expreffing the number of Chances required. Thus if it be required to affign the number of Chances for throw- ing precifely three Aces, with ten Dice ; here /will be = 3, and n ■== \o. Now becaufe / is lefs than -^ «, let fo many terms be taken of the Series &c. as there are Units in 3, which terms in this particular cafe will be — , 1 — • 1 2 ' 3 ' let thofe terms be multiplied together, the produdl will be 120 j let this product be again multiplied by a'b''—', that is {a being =: i, ^ = 9, /= 3, « = 10) by 6042969, and the new produd: will be 725156280, which confequently exhibits the number of Chances required. Now this being divided by the 10'" power of ^ -\- b, that is, in this cafe, by looocoooooo, the quotient 0.0725156280 will cxprefs the Probability of throwing precifely three Aces with ten Dice ; and this being fubtradled from Unity, the remainder 0.9274843720 will exprefs the Probability of the contrary; and therefore the Odds againft throwing three Aces precifely with ten Dice are 9274843720, to 725156280, or nearly as 64 to 5. Although we have fhewn above how to determine univerfally the Odds of winning, when two Adverfaries being at play, refpedtively want certain number of Games of being up, and that they have any given proportion of Chances for winning any fingle Game ; yet I have thought it not improper here to annex a fmall Table, (hew- ing thofe Odds, when the number of Games wanting, does not exceed fix, and that the Skill of the Contenders is equal. Games Odds of Games Odds of Games Odds of wanting. winning. wanting. winning. wanting, winning. I, 2 - - - .^> J 2. 3 - - ^^ 5 3, 5 - - 99, 29 I' 3 - - - 7. I 2, 4 - - 26, 6 3, 6 - - 219, 37 I, 4 - - - M. ' 2, 5 - - 57^ 7 4. 5 - - 163, 93 ^ 5 -- - 3^ I 2, 6 - - 120, 8 4, 6 - - 382, 130 I, 6-- - ft'' 1 3' 4 - - 42. 22 5, 6 - - 638, 386 Before The Doctrine^?/" Chances. 29 Before I put an end to this Introdudlion, it will not be impro- per to fhew how fome operations may often be contradted by barely introducing one fingle Letter, inftead of two or three, to denote the Probability of the happening of one Event. 18. Let therefore x denote the Probability of one Event; j, the Probability of a fecond Event ; ^, the Probability of the happening of a third Event : then it will follow, from what has been faid in the beginning of this Introduction, that i — x, i — y\ i — z will reprefent the refpe<flive Probabilities of their failing. This being laid down, it will be eafy to anfwer the Queflions of Chance that may arife concerning thofe Events. 1°. Let it be demanded, what is the Probability of the happening of them all ; it is plain by what has been demonftrated before, that the anfwer will be denoted by xyz. 2°. If it is inquired, what will be the Probability of their all failing ; the anfwer will bei — x y. i — y y. j — 2;, which being expanded by the Rules of Multiplication would be i — x — y — z -f- xy -\- xz -\- yz — xyz ; but the firfl expreffion is more eafily adapted to Numbers. 3'. Let it be required to aflign the Probability of fome one of them or more happening ; as this queftion is exaftly equivalent to this other, what is the Probability of their not all failing ? the anfwer will be I — 1 — XX I — y X 1 — 2;, which being expanded will become x -\- y -\- z — xy — xz ■ — • yz -\- xyz. 4°. Let it be demanded what is the Probability of the happening of the firft and fecond, and at the fame time of the failing of the third, the Queftion is anfwered by barely writing it down algebrai- cally ; thus, xy X I — z, or xy — xym : and for the fame reafon the Probability ofthe happening ofthe firft and third, and the failing of the fecond, will he xz x 1 — ^ or xy — xyz : and for the fame reafon again, the Probability of the happening of the fecond and third, and the failing of the firft, will be yz x i — x, or yz — xyz. And the Sum of thofe three Probabilities, viz. xy -\- xz -\- yz — 3.y)'2:. will exprefs the Probability of the happening of any two of them, but of no more than two. 5°. If it be demanded what is the Probability of the happening of the fiift , to the exclufion of the other two, the anfwer will be X X I — y XI — z, or X — xy — xz -]- xyz ; and in the fame man- ner, the Probability of the happening of the fecond to the exclu- fion of the other two, will be ^ — xy — yz -f- xyz ; and again,^ the Probability of the happening of the third, to the exclufion of the. other 3© 'Jthe Doctrine o/" Chances. other two, will be z — xz — yz •\- xyz, and the Sum of all thefc Probabilities together, liiz. x -\- y -\- z — 2xy — 2xz — 2yz •\- T,xyz will exprefs the Probability of the happening of any one of them, and of the failing of the other two : and innumerable cafes of the fame nature, belonging to any number of Events, may be folved without any manner of trouble to the imagination, by the mere force of a proper Notation. Remark. I, When it is required to fum up fevcral Terms of a high Power of the Binomial a -\- b, and to divide their Sum by that Power, it will be convenient to write i and q for a and b; having taken q'. 1 : : b '.a '. and to ufe a Table of Logarithms. As in the Example of Art. ly^^, where we had to compute '"^'^t + ^^r' ' ^ ^^^"S — - 3. ^ = 5 > we fhallhave y = i- , and, inftead of the former, we are now to compute the quantity Now the Fador q^ -\- gq ^ 36 being ^ -}- ^ + 36 = i?i , Whofe Logarithm is L. 484 — L. 9 r= - - 1,7306029 Add the Log. of q'^, or y xL. 5 ~ L. ^ = - - i.5<^29409 And from the Sum - - - - 3-2835438 Subtradl the Log. of 7*4^ '> or 9 x L.2 — L. 3 = 3.8337183 So ihall the Remainder - - , - - — 1.4498255 be the Logarithm of B's chance, viz. 0.281725 And the Complement of this to Unity 0.718275 is the Chance of A, in that Problem of Jrt. 1 7'''. An Operation of this kind will ferve in moft cafes that occur : but if the Power is very high, and the number of terms to be fummed cxceffively great, we muft have recourfe to other Rules -, which fhall be given hereafter. IL When the Ratio of Chances, which we (hall call that of R to S, comes out in larger numbers than we have occafion for ; it may be reduced to its leajl exaSleJi Terms, in the Method propofed by Dr. Wallis, Huygens, and others. As thus ; Divide the greater Term jR by the lefler S ; the lafl: Divifor by the Remainder -, and fo on continually, as in finding a common Divifor : and let the feveral Quotients, in the order they arife, be reprefented 7/5^ Doctrine <9/" Chances. 31 reprefented by the Letters a, b, c, d, e, &c. Then the Ratio -^ , of the lefler Term to the greater, will be contained in this fradion- al Series. d-^x e + isc. whofe Terms, from the beginning, being reduced to one Fraflion, will perpetually approach to the juft Value of the Ratio -^> differ- ing from it in excefs and in defeSi, alternately : fo that if you flop at a Denominator that ftands in the i'', 3'', 5'\ Off. place j as at a, c, e, &c. the Refult of the Terms will exceed the juft Value of the Ratio -^ ; but if you ftop at an even place, as at ^, d^ /, &c. it will fall fhort of it. Example I. If it is required to reduce the Ratio juft now found -frg^ » or i|^ ^ = — j to lower Terms -, and which (hall exhibit its juft Quantity the neareft that is poffible in Terms fo low : The Quotients, found as above, will bej a =2, b^i, c = i, ^=4, <?^ ij/=i,^ = 5- And, i'. The firft Term -1 , or — , gives the Ratio too great ; becaufe its confequent a is too little. 2°. The Refult of the two firft Terms 7:^7 = ^^ = j . is lefs T T than ^ , altho' it comes nearer it than -^ did : becaufe j = r, which we added to the Denominator 2, exceeds its juft Quantity 6+ i_ c+(sfc. 5°. The three firft Terms ^^ = ~-^ ; which reduced are C I j^ = ~ exceeds the Ratio j--, becaufe what we added tt> 2 32 7^^ Doctrine ^ Chances. the Denominator b exceeding its juft Quantity f^ calces jqp2 too little, and confequently the whole Fradion too great. c In the fame manner, the following Approximation Q 7 ^1 — 7+ 7 = Tj > tho' juftcr than the pre- ^-j- 1 1 + I d . ^ ceding, errs a little in defecft. And fo of the reft. But to fave unneceflary trouble ; and to prevent any miftake ei- ther in the Operation itfelf, or in diftinguIlLing the Ratios that ex- ceed or fall fhort of their juft Quantity ; we may ufe Mr. Cotes's Rule ; which is to this purpofe. Write S : R2Li the head of two Columns, under the T'ltXes greater, and lefs. And place under them the two firft Ratios that are found ; as in our Example 1:2, and 1:3- Multiply the Terms of this laft Ratio by the third Denominator f, and write the Products under the Terms of the firft Ratio 1:2. So £hall the Sums of the Antecedents and Confequents give a jufter Ratio 2 : 5, belonging to the left-hand Column. Multiply the terms of this laft by the 4"' Quotient d (= 4), and the Produds added to I : 3 give the Ratio 9:23, belonging to the right-hand Co- lumn. This laft multiplied by e (= i), and the Produds transfer- red to the left hand Column, and added to the Ratio that ftood laft there, give the Ratio 11 : 28. And fo of the reft, as in the Scheme below. greater S : R S lefs : R i-— I : a I : 2 3 I • 3 ^ i , X c 1 ^7 2 : 5> 'd — 4 8 : 20 9 : 23 2«X f I 9 1 1 : 23 X f I : 28 100 : 25? 20 '' S» Xg — s III ; 283 &c. This Method is particularly ufeful, when furd numbers, which h^ve no Termination at all, enter into any Solution. Example Tie Doctrine e/" Chances, 33 Example II. It will be found In the Refolution of our firft Problem that the proportion of Chances there inquired into ( j) is that of i to Va — i^ or of 1 to 0.259921 &c. Whence our Quotients will bej tf = j» ^=j, <:=5, J=i, ^ = i,/=4, &c. And the Operation will ftand as below. greater S : R lefs S : R -=1:3 < : 20 X f 5 * 6 : 23 X</— . I 6 : 23 7 : 27 7 : 27 X ^ = 1 J3 : 50 x/ 4 52 : 200 Off. 59 : 227 End of the IntroduSiion^ Solutions 35 Solutions of feveral forts of Problems, deduced from the Rules laid down in the Introdudion. PROBLEM I. Jf A and B play isoith fingh Bowls y and fuch be the Jkill of A that he knows by Experience he can give B two Games out of three ; what is the proportion of their Jkill J or what are the Odds^ that A may get any one Game ajfgnedf Solution. ET the proportion of Odds be as z to i ; now fince A can give B 2 Games out of 3, y^ therefore may upon an equality of Play undertake to win 3 Games together : but the probability of his winning the ikft time is -—7-, and, by the %th Article of the IntroduElioriy. the probability of his winning three times together is -^ — x ? x- ^ ? , or _^ ,, . Again, becaufe A and E are fuppofed to play upon equal terms, the probability which A has of winning three times together ought to be expreffed by — ; we have therefore the Equation ■ = ^ , otzz^^=.z -f-i^', and extrading the cube- root on both fides, z^i -^z -\- \\ wherefore s = , andcon- fequently the Odds that A may get any one Game affigned are as to I, or as I to >/2 — i, that is in this cafe as 50 to 13 ve- ry ncar^ F 2 Corol- 36 The Doctrine <?/ Chances. Corollary. By the fame procefs of inveftigation as that which has been ufcd in this Problem, it will be found that if A can, upon an equality of Chance, undertake to win n times together, then he may juftly lay n the Odds of I to V 2 — i, that he wins any one Game afligned. PROBLEM II. If ^ can without advajttage or difadvantage give B i Game out of 3 ; what are the Odds that A poall take a?ty one Game ajftgnedf Or in other terms ^ what is the proportion of the Chances they refpeSiively have of win- ning any one Game ajjigned f Or what is the propor- tion of their Jkillf Solution. Let the proportion be as 2; to i : and llnce A can give B i Game out of 3 ; therefore A can upon an equality of play undertake to win 3 Games before J5 gets 2: now it appears, by the ly'*" Art, of the LitroduSlion, that in this cafe the Binomial z '\- \ ought to be raifed to its fourth power, which will be z'^-\-/[.z^ -}- 6zz -}- 45; -}- i j and that the Expedlation of the firft will be to the Expectation of the fecond, as the two firft terms to the three laft : but thefe £x- pe(flations are equal by hypothefis, therefore 2* -f- 423 -^^z tzz -\- i^z-\- i: which Equation being folved, z will be found to be 1.6 very near; wherefore the proportion required will be as 1.6 to i, or 8 to 5. PROBLEM in. To find in how many Trials an Event will probably happen^ or how many Trials will be necejfary to make it indiffe- rent to lay on its Happening or Failing ; fuppoftng that a is the number of Chances for its happening in any o?ie Trialy and b the 7 lumber of Chances for its failing. Solution. Let X be the number of Trials; then by the 16'* Art. of the Jntrod. b" will reprcfent the number of Chances for the Event to fail X times fucceflively, and a-\-b) " the whole number of Chances for happen- *The Doctrine ^Chances. 37 happening or failing, and therefore . reprefents the probabi- lity of the Event's falling x times together: but by fuppofition that Probability is equal to the probability of Its happening once at lead in that number of Trials ; wherefore either of thofe two Pro- babilities may be exprefTed by the fradion — : we have therefore the Equation =- =f , or a -\- bY = 2b\ from whence is de- duced the Equation x log. a ^- b = x log. b ■\- log. 2 ; and therefore Log. a •\- b — log. b Moreover, let us reaflume the Equation "7+7)' = 2^', where- in let us fuppofc that a, b : : i, y ; hence the faid Eq uation will be changed into this i + -"*' =2. Or at x log. 14--^= log. a. In this Equation, if y be equal to i, x will likewife be eq ual to i ; but if q differs from Unity, let us in the room of log. i -}- - write its value expreffed In a Series ; viz. I -A-^ L + J L. &c We have therefore the Equation — — — , 6cc. = log. 2. Let us now fuppofe that q is infinite, or pretty large in refpeft to Vnity^ and then the firft term of the Series will be fufficient; we fhall therefore have the Equation-^ = log. 2, or x = y log. 2. But it Is to be obferved in this place that the Hyperbolic, not the Ta- bular, Logarithm of 2, ought to be taken, which being 0.693, &c. or 0.7 nearly, it follows that x = oqq nearly. Thus we have afligned the very narrow limits within which the ratio of >: to ^ is comprehended ; for it begins with unity, and ter- minates at laft in the ratio of 7 to 10 very near. But X foon converges to the limit 0.75', fo that this value of x may be affumed in all cafes, let the value of q be what It will. Some ufes of this Problem will appear by the following Examples. Example i. hct it be propofed to find in how viany throias one may undertake with an equality oj Chance, to throiv two Aces with two Dice. The number of Chances upon two Dice being 36, out of which there is but one chance for two Aces, it follows that the number of Chances 38 7T)e Doctrine of Chances. Chances agalnft it is 35; multiply therefore 3 5 by 0.7, and the produd 24.5 will fhew that the number of throws requifite to that effed will be between 24 and 25. Example 2. To find in how many throws of three Dice, one may undertake to throw three Aces. The number of all the Chances upon three Dice being 216, out of which there is but one Chance for 3 Aces, and 2 1 5 againft it, it follows that 215 ought to be multiplied by 0.7; which being done, the produdl 150.5 will fhew that the number of Throws requifite to that effed will be 150, or very near it. Example 3. In a Lottery whereof the number of blanks is to the number of prizes as 2,^ to I, {fuch as was the Lottery in 17 10) to find how many Tickets one mufl take to make it an equal Chance for one or more Prizes. Multiply 39 by 0.7, and the produdl 27.3 will fhew that the num- ber of Tickets requifite to that efFedl will be 27 or 28 at mofl. Likewife in a Lottery whereof the number of Blanks is to the number of Prizes as 5 to i, multiply 5 by 0.7, and the produdt 3.5 will fhew that there is more than an equality of Chance in 4 Tickets for one or more Prizes, but lefs than an equality in three. Remark. In a Lottery whereof the Blanks are to the Prizes as 39 to i, if the number of Tickets in all were but 40, the proportion above- mentioned would be altered, for 20 Tickets would be a fufficient number for the juft Expectation of the fingle Prize ; it being evi- dent that the Prize may be as well among the Tickets which are taken, as among thofe that are left behind. Again if the number of Tickets in all were 80, ilill prefervlng the proportion of 39 Blanks to one Prize, and confequently fup- pofing 7* Blanks to 2 Prizes, this proportion would flill be altered;. for by the Dodlrine of Combinations, whereof we are to treat after- wards, it will appear that the Probability of taking one Prize or both in 20 Tickets would be but ^ , and the Probability of ta- king none would be ^ ; wherefore the Odds againft taking any ^rize would be as 177 to 139, or very near as 9 to 7» And Use Doctrine o/'Chances. 39 And by the fame Dcxftrine of Combinations, it will^be found that 23 Tickets would not be quite fufhcient for the Expedation of a Prize in this Lottery; but that 24 would rather be too many: fo that one might with advantage lay an even Wager of taking a Prize in 24 Tickets. If the proportion of 39 to i be oftner repeated, the number of Tickets requisite for the equal Chance of a Prize, will flill increafe with that repetition; yet let the proportion of 39 to i be repeated never fo many times, nay an infinite number of times, the number of Tickets requifile for the equal Chance of a Prize would never exceed — of 39, that is about 27 or 28. Wherefore if the proportion of the Blanks to the Prizes is often repeated, as it ufually is in Lotteries ; the number of Tickets requifite for a Prize will always be found by taking — of the proportion of the Blanks to the Prizes. Now in order to have a greater variety of Examples to try this Rule by, I have thought fit here to annex a Lemma by me pub- lifhed for the firft time in the year 171 1, and of which the in- veftigation for particular reafons was deferred till I gave it in my Mifcellanea Analytica anno 1 73 1. Lemma. ' To find bow many Chances there are upon any number of Dice, each of them of the fame number of Faces, to throw any given number of points. Solution. Let^ -f- I be the number of points given, n the number of Dice, y the number of Faces in each Die : make /» — /= q, q — f=^ r, r — y = J> i — /==^> &c. and the number of Chances required will be + f "■^-''V'^^'^- — X 1 X , &C. I 2 3 » n X — I 1 r r — I r — 2 g X X , &C. 12 3 ' j» n — I X — X 1 Z - ; X ';■ X '-- , &c. n t> — I X— X I 2 » 2 X 3 -i-&c. Which Series's ought to be continued till fome of the Fadors in each produdl become either = o, or negative. ^ N.B. 40 'The Doctrine o/* Chances. N. B. So many Fadlors are to be taken in each of the produdls — X .^^ x-^— ^, &c. — X -^^^ X -^^ , &c. as there are Units in « — 1. Thus for Example, let it be required to find how many Chances there are for throwing 1 6 Points with four Dice ; then making p-\- 1 = i6, we have />= 15, from whence the number of Chances re- quired will be found to be +-r«^^-f = + 455 08-74 - — -2x-x- X— =r — ^•?6 123 I •^•^ + 1 xi. X ± X 1 X 1 = 4- 6 But 455 — 336-^6=125, and therefore one hundred and twenty- five is the number of Chances required. Again, let it be required to find the number of Chances for throwing feven and twenty Points with fix Dice ; the operation will be A.JLy,2L^J± x^x-iL =4-65780 T^ I 2 3 4 s I Jf 20 iq 18 17 16 6 ^^^^ . i_x— -X — X— i- X — X— = — 93024. I 2 3 4 5 I yj T . ^x-^x-Ii-x-LLx-Ilxl x-^ =+30030 87 6? 465 4 X- X— x-l X- X— X- x-=: — 1 120 12 3 4 5 ' 2 3 Wherefore 65780 — 93024 -\- 30030 — 1 120 = 1666 is the num- ber of Chances required. Let it be farther required to afllgn the number of Chances for throwing fifteen Points with fix Dice. JUJi-x— -X — X — X — =4- 2002 '12 3 4 5 ' __lx^xixi-x±xl=-336 I 2 3 4 5 « ^^ But 2002 336 = 1666 which is the number required. Corollary. All the points equally diftant from the Extremes, that is from the lead and greatefl: number of Points that are upon the Dice, have the fame number of Chances by which they may be produ- ced ; wherefore if the number of points given be nearer to the greater Extreme than to the lefi"er, let the number of points given be The Doctrine o/* Chances. 41 be fubtraded from the Sum of the Extremes, and work with the re- mainder ; by which means the Operation will be Shortened. Thus if it be required' to find the number of Chances for throw- ing 27 Points with 6 Dice: let 27 be fubtradtcd from 42, Sum of the Extremes 6 and 36, and the remainder being 15, it maybe concluded that the number of Chances for throwing 27 Points is the fame as for throwing 1 5 Points. Although, as I have faid before, the Demonftration of this Lemma may be had from my Mifcellanea ; yet I have thought fit, at the defire of fome Friends, to transfer it to this place. Demonstration. 1°. Let us imagine a Die fo conftituted as that there (hall be upon it one fingle Face marked i , then as many Faces marked 1 1 as there are Units in r, and as many Faces marked 1 1 1 as there are Units in rr, and fo on ; that the geometric Progreffion i -\- r -\- rr -\- r'^ -\- r+ -\- r^ -\- r^ Ar r'' -\- r^ 6cc. continued to fo many Terms as there are different Denominations in the Die, may reprefent all the Chances of one Die : this being fuppofed, it is very plain that in order to have all the Chances of two fuch Dice, this ProgrefTion ought to be raifed to its Square, and that to have all the Chances of three Dice, the fame ProgrefTion ought to be raifed to its Cube ; and univerfally, that if the number of Dice be exprefled by «, that Progreflion ought accordingly to be raifed to the Power ;z. Now fuppofe the number of Faces in each Die to be f, then the Sum of that Pro- — / greflion will be '_^ - j and confequently every Chance that can happen upon n Dice, will be exprefled by fome Term of the Series _ / that refults from the Fradlion |_^ raifed to the power n. But as the leafl number of Points, that can be thrown with n Dice, is n Units, and the next greater n-\-i, and the next Ji -\- 2, Sec. it is plain that the firfl: Term of the Series will reprefent the number of Chances for throwing « Points, and that the fecond Term of the Se- ries will reprefent the number of Chances for throwing « -\~ i Points, and fo en. And that therefore if the number of Points to be thrown be exprefl"ed hy p -\- j, it will be but afTjgning that Term in the Series of which the diflance from the firft Ihall be exprelTed by p 4- I — «. G But 42 "The Doc TRINE (?/'Chances. But the Series which would refult from the raifing of the Fradllon _/ - — - to the Power «, is the Produdl of two other Series, whereof I — r ' i" "4-1 l" "4-' "+2,0 1 one IS I 4- ;/r -|- 7 X -^— rr -\ - y. —~ x — — r^, &c. the Other IS i — «r/ + - x — — r'f -- - x — — x r ■/ + &c. Wherefore, if thefe two Series be multiplied together, all the Terms of the produd will feverally anfwer the feveral numbers of Chances that are upon n Dice, And therefore if the number of Points to be thrown be exprefled by ^4- I, it is but colledling all the Terms which are affedted by the Power r*+'~", and the Sum of thofe Terms will anfwer the Queftion propofed. But in order to find readily all the Terms which are affeded by the Power r^+'~", let us fuppofe, for fhortnefs fake, p -\- i — n=z/; and let us fuppofe farther that Er' is that Term, in the firfl Series, of which the diflance from its firfl; Term is /; let alfo Dr'—f be that Term, in the firfl Series, of which the diflance from its firft Term is / — f, and likewife let Cr'—^f be that Term, in the firfl Series, of which the diflance from its firfl Term is denoted by / — 2/, and fo on, making perpetually a regrefs towards the firfl Term. This being laid down, let us write all thofe Terms in order, thus • Eri -\- V>r'~f + Cri-^f + Br'- if, &c. and write underneath the Terms of the fecond Series, in their natu- ral order. Thus Eri -\- Dr'-f + Cr-'—^f -\- Br'-?/, &c. I —nrf + 7 '^ -^'-f + 7 X -7- X -j-rV, Sec. then multiplying each Term of the firfl Series by -each correfponding Term of the fecond, all the Terms of the produdl, viz. Erl - nDr' _l- JL x ^l^i- Cr' - - x ^Hl- x lZl.Br', Sec. '12 12 3 will be affefted with the fame power r'- Now the Coefficient E containing fo many favors 7 x L-L x ■"+^ &c. as there are Units in/j it is plain, that when the De- nominators of thofe fadors are continued beyond a certain number of them, denominated by « — i, then the following Denominators will be «, n-\- i, n-\-2, &c. which being the fame as the firfl Terms of the Numerators, it follows that if from the value of the Coefficient E be rejeded thofe Numerators and Denominators which are The Doctrine ^Chances. 43 are equal, there will remain out of the Numerators, written in an inverted order, the Terms n -\-l — i, n ■\- 1 — 2, n-\- /— 3, &c. of which the laft will be / -|- i ; and that, out of the Denomi- nators written in their natural order, there will remain i, 2, 3, 4, 5, &c. of which the laft will be « — i : all which things depend intirely on the nature of an Arithmetic Progrelhon. Wherefore the firfl Term Er' IS = ; X J— X ^ _±_K Now in the room of /, fubftitute its value ^ -j- 1 — », then Er' ^=z — X -^p- X — -^ , &c. X r^, and in the fame manner will the lecond Term — «Dr-' be =— i:=^ X ^~^~' X f'^-' &c. x nr', and alfo the third Term 4. ^ X iml Cr/ will be = -1- ±=^ X Iz:±zL x £:::l£^ &c. X ~ X -l^^r/, and fo on. Suppofe now r = i , ^ — f^^q^q — f=:.r^ r — /"= i, &c. and you fhall have the very Rule given in our Lemma. Now to add one Example more to our third Problem, let it be required to find in how many throws of 6 Dice one may undertake to throw 15 Points precifely. The number of Chances for throwing 15 Points being 1666, and the whole number of Chances upon 6 Dice being 46656, it fol- lows that the number of Chances for failing is 44990 j wherefore dividing 44990 by 1666, and the quotient being 27 nearly, multi- ply 27 by 0.7, and the produdl 18,9 will fhew that the number of throws requifite to that eflfcdl will be very near 19. PROBLEM IV. To find how many Trials are necejfary to make it equally probable that an Event will happen twice^ /'■^PP^^fi^^g, that a is the number of Chances for its happenino- in any one Trials and b the munber of Chances for its faili?ig. Solution. Let X be the number of Trials: then from what has been demon- flrated in the i6'h jirt. of the Introd. it follows that b^ -j- xab^-' is G 2 the 44 Th& Doctrine ^Chances. the number of Chances whereby the Event may fail, a -|- /> V' com- prehending the whole number of Chances whereby it may either happen or fail, and confequently the probability of its failing is .: but, by Hypothefis, the Probabilities of happening and failing are equal ; we have therefore the Equation //- a+n X 2 , oio-\-b\'' = 2b'' + zxab^'-^y or making a, b :: i. I -|" q'''^^^'^~T' ^°w if in this Equation we fuppofe qz=.ij X will be found =3, and if we fuppofe q infinite, and alfo - =: 2;, we fhall have the Equation z = log. 2 -\- log. i -\- z, q in which taking the value of z, either by Trial or othcrwife, it will be found = 1.678 nearly ; and therefore the value of x will always be between the limits iq and 1.6785', but will foon converge to the laft of thefe limits ; for which reafon, if q be not very fmall, x may in all cafes be fuppofed = 1.678^; yet if there be any fufpicion that the value of X thus taken is too little, fubflitute this value in the original Equation 1 -\- -*"'=2+ — , and note the Error. Then if it be worth taking notice of, increafe a little the value of x, and fub- flitute again this new value of x in the aforefaid Equation ; and noting the new Error, the value of x may be fufficiently corrected by applying the Rule which the Arithmeticians call double falfe Pofition. Exam p l e i. T!o find in how many throws of three Dice one may undertake to throw three Aces twice. The number of all the Chances upon three Dice being 216, out of which there is but 1 Chance for three Aces, and 215, againft it ; multiply 215 by 1.678 and the produdl 360.8 will (hew that the number of throws requifite to that efFea: will be 361, or very near it. Example 2. to find in how many throws of 6 Dice one may undertake to throw 15 Points twice. The number of Chances for throwing 15 Points is 1666, the number of Chances for miffing 44990 j let 44990 be divided by 1666, 'The Doctrine 0/ Chances. 45 1666, the Quotient will be 27 very near : wherefore the Chances for throwing and miffing 15 Points are as i to 27 refpedlively j mul- tiply therefore 27 by 1.678, and the produdl 45.3 will (hew that the number of Chances requifue to that effed will be 45 nearly. Example 3. In a Lottery lohercoj the nwnber of Blanh is to the Number of Prizes as 29 to I : to Jind how many "ticket a niujl be taken to make it probable that two or more benefits will be taken as not. as Multiply 39 by 1.678 and the produdt 65.4 will fhew that no lefs than 65 Tickets will be requifite to that effedt. PROBLEM V. To Jind how ma?iy Trials are necejfary to make it equally probable that an Rve7it will happen three^ four^ fivey &'c. times ; fuppojing that a is the number of Cha?ices for its happening in any one Trialy a?jd b the 7tumber of Chances for its failing. Solution, Let X be the number of Trials requifite, then fuppofing as be- fore a, b : : J , ff, we fhall have the Equation i -\- lA' = 2 x 1 -\- — -\- — X -ill^ , in the cafe of the triple Event ; or + l\X I X , X X — 1 , X X 1 ,, x—z the cafe of the quadruple Event : and the law of the continuation of thefe Equations is manifefl. Now in the firft Equation if g be- fuppofed = I, then will at be = 5; if ^ be fuppofed infinite or pretty large in refpeft to Unity, then the aforefaid Equation, mak- ing — = 2;, will be changed into this, z = log. 2 -\- log. I -\- z -^ — zz ; vv^herein z will be found nearly 7= 2.675", where- fore X will always be between ^q and 2.675^. Likewife in the fecond Equation, if q be fuppofed = i, then will X he = jq; but if q be fuppofed infinite or pretty large in re- ^eft to Unity, then z = log. 2 -{- log, 1 -\- z -{- j zz + f z^; whence. 46 T^he Doctrine 0/ Chances. ■whence z will be found nearly = 3 6719, wherefore x will be be- tween 7J' and 3.6719^. A Table of the Limits. The Value of x will always be For a fingle Event, between \q and 0.693^ For a double Event, between 3^ and i.ty'iq For a triple Event, Lt:ween 5^ and 2.6757 Fora quadruple Event, between jq and 3.6727 For a quintuple Event, between gq and 4670^ For a iextuple Event, between i iq and 5.668^ &c. And if the number of Events contended for, as well as the num- ber q be pretty large in refped: to Unity ; the number of Trials requifite for thofe Events to happen n times wid be ^"~^ q^ or barely nq. Remark. From what has been faid we may plainly perceive that altho' we may, with an equality of Chance, contend about the happening of an Event once in a certain number of Trials, yet we cannot, without difadvantage, contend for its happening twice in double that number of Trials, or three times in triple that number, and fo on. Thus, altho' it be an equal Chance, or rather more than an Equa- lity, that I throw two Aces with two Dice in 25 throws, yet I can- not undertake that the two Aces flrall come up twice in 50 throws, the number vequifite for it being 58 or 59 ; much lefs can I under- take that they (hall come up three times in 75 throws, the number requifite for it being between 93 and 94 : fo that the Odds againft the happening of two Aces in the iirft throw being 35 to i, I cannot un- dertake that in a very great number of Trial?, the happening fliall be oftner than in the proportion of i to 35. And therefore we may lay down this general Maxim, that Events at long run will not hap- pen oftner than in the proportion of the Chances they have to hap- pen in any one Trial ; and that if we afTign any other proportion varying never fo little from that, the Odds againft us will increafe continually. To this may be objedted, that from the premifes it would feem to follow, that if two equal Gamefters were to play together for a coiifiderable time, they would part without Gain or Lofs on either fide : but the anfwer is eafy ; the longer they play the greater Pro- bability The Doctrine <?/'Chances. 47 bability there is of an increafe of abfolute Gain or Lofs ; but at the fame time, the greater Probability there is alfo of a decreafe, in refpedt to the number of Games played. Thus if 100 Games produce a dif- ference of 4 in the winnings or lofings, and 200 Games produce a difference of 6, there will be a greater proportion of Equality in the fecond cafe than in the firft. PROBLEM VI. Th-ee Gamejlers A, B, C flay together on this conditiotir that he Jloall ivin the Set who has Joonejl got a certain number of Games ; the proportion of the Chances ii-hich each of them has to get a7iy one Game ajjlgned, or which is the fa?ne things the proportion of their Jkill^ being re- JpeSiively as a, b, c. Now after they have played foi}ie time^ they fnd the7?ifelves in this circumfiance^ that A wants I Game of being up^ B 2 Games^ a?id C 3 Gajnes ; the whole Stake a?no?igJl them being fuppofed i ; what is the Expe&ation of each ^ Solution. I. In the circumftance the Gamefters are in, the Set will be ended in 4 Games at moft ; let therefore a-\- b •\- ch& raifed to the fourth power, wliich will be a'^ •\- ^a'^b -\- 6aabb -\- ^ab^ -|- b^ -\- ^a'^c -\- izaabc -j- i^b c -\- baacc -\- \2abcc -\- tbbcc -j- ^ac'^ -\- izacbb -j- 4^c3 -f c\ The terms a* ■\- i^a^b -\- ^a^c 4" ^aacc -\- xiaabc ■\- \2abcc, wherein the dimenfions of a are equal to or greater than the number of Games which A wants, wherein alfo the Dimenfions of b and c are lefs than the number of Games which B and C refpeitively want, are intirely favourable to A, and are part of the Numerator of his Expedition. In the fame manner, the terms b'' -\- ^b^c -{- bbbcc are intirely fa- vourable to B. And likewife the terms /^bc^ -\- c^ are intirely favourable to C. The reft of the terms are common, as favouring partly one of the Gamefters, partly one or both of the other ; wherefore thcfc Terms are fo to be divided into their parts, that the parts, refpedively fevouring each Gamefter, may be added to his Expedition. Take 48 ^^ Doctrine 0/ Chances. Take therefore all the terms which are common, viz. 6aabb, ^ab^, izabcc, ^ac"^, and divide them adually into their parts ; that is, 1°, taabb into aabb, abab, abba, baab, baba, bbaa. Out ot thefe fix parts, one part only, -viz. bbaa will be found to favour £, for 'tis only in this term that two Dimenfions of b arc placed before one fingle Dimenfion of a, and therefore the other five parts belong to A ; let therefore '^aabb be added to the Expetflation of A, and \aabb to the Expectation of B. 2°. Divide i^ab^, into its parts ^bbb, babb, bbab, bbba ; of thefe parts there are two belonging to A, and the other two to B; let therefore zab^ be added to the ex- pectation of each. 3". Divide i labbc into its parts ; and eight of them will belong to A, and 4 to fi ; let therefore ?iabbc be added to the Expedation of A, and i,abbc to the Expectation of B. 4". Di- vide 4^6-3 into its paj-ts, three of which will be found to be favourable to A, and one to C; add therefore ^ac^ to the Expectation of A, and Mc5 to the Expectation of C. Hence the Numerators of the feveral lixpeCtationsofy^, B, C, will be refpeCtively, j^ ^+ j|_ ^a^l; -\- ^a''C -\- 6aacc -\- iiaabc -\- iiabcc -\- ^aabb -I- 2ab^ + ^abbc -f ^acK 2. i+ J^ j^b^c + (:>bbcc -\- laabb -\- 2ab> + ^obcc. 3. 4tbc'^ -]- iC^ -\- iac^- The common Denominator of all their Expectations hdnga-\-b-{-c)*. Wherefore if a, b, c, arc in a proportion of equality, the Odds of winning will be refpeCtively as 57, 18, 6, or as 19, 6, 2. If n be the number of all the Games that are wanting, p the number of Gamefters, and a, b, c, d, &c. the proportion of the Chances which each Gamefter has refpeCtively to win any one Game afligned ; kt a -\- b -{- c -\- d, &c. be raifed to the power « -\- 1 — />, and then proceed as before. Remark. This is one general Method of Solution. But the fimpler and more common Cafes may be managed with very little trouble. As, 1°. Let A and B want one game each, and C two games. Then the following game will either put him in the fame fituation as A and B, entitling him to -^ of the Stake ; of which there is i Chance ; or will give the whole Stake X.o Aor B ; and of this there iX-^+zXo are two Chances. Cs Expectation therefore is worth — ^- (hitrod. 7"^^ Doctrine o/ Chances. 49 (Introl Art. 5.) =-• '^^^^ ^^^"'^ ^^^^ ^^^ ^^^^^ '' ^"^ ^^^ ^^' mainder - , to be divided equally between A and B, makes the ex- pedations oi A, B, C, to be 4, 4, i, refpedively ; to the common Denominator 9. 2°. Let ^ want i Game, B and C two games each. Then the next Game will either give A the whole Stake ; or, one of his Ad. verfaries winning, will reduce him to the Expedlation - , or the former Cafe. His prefent Expedation therefore is — 2 =—: and the Complement of this to Unity, -viz. -^ , divided equally between Band C, gives the three Expectations, 17, 5, 5, the com- mon Denominator being 27. 3°. A and B wanting each a Game, let C want 3. In this Cafe, C has 2 Chances for o, and i Chance for the Expedation — , of Cafe I. That is, his Expedtation is -^ ; and thofe of yf and B are-j^ , each. 4°. Let the Games wanting to A, B, and C, be i, 2, 3, re- fpedively : then A winning gets the Stake i ; B winning, A is in Cafe 3, with the Expedlation -^ , or C winning, he has, as in Cafe 2, the Expedation -^. Whence his prefent Expedlation is 27 2. X I -I- -^ _L. _L_ :=, -!_ 3 ' ;- ' 27 «i • Again, ^ winning, B gets o ; himfelf winning, he acquires {Cafe 3.) the Expedlation -^ . And, C winning, he is in Cafe 2, with the Expedlation -^ . His prefent Expedlation therefore is 4" >< o -l-4r "1" ~^ = 7r- ^'^'^ ^^is to the Expedlation of ^, which was —^i the Sum is -^ : and the Complement of this to Unity, which is -^ , is the Expedlation of C. Or to find Cs Expedlation diredlly : A winning, C has o ; B win- ning, he has the Expedlation -;^ , {Cafe 3.) and, himfelf winning, he has-^ , as in Cafe 2 : In all, ^ x o + '77"Hi7= TT' H And 50 "The Doctrine <?/ Chances. And thus, afcending gradually through all the inferior Cafes, or by the general Rule, we may compofe a Table of Odds for 3 Gamfters, fuppofed of equal Skill $ like that for 2 Gamefters in Art, if'^ of the Introdudion. Table for 3 Gamejlers. Games wanting. Odds. Games wanting. Odds. Games wanting. Odds. A. B. C a. b. c A B. C. a. b. c. ^A. B. C. rt. ^. f. 1 I 2 ' ' 3 1 1 4 I I 5 I 2 2 I 3 3 4 4 I 13 '3 1 40 40 I 121 121 ] \1 5 5 65 8 8 I 2 3 I 2 4 I 2 5 I 3 4 1 3 5 2 2 3 1962 178 58 7 542 179 ^ 616 82 31 629 87 13 34 34 ^l 224 2 2 5 233 234 2 3 5 33« 33« 53 353 353 23 133 55 55 451 195 83 H33 635 119 &c. Solution II. and more General. It having been objecfled to the foregoing Solution, that when there are feveral Gamefters, and the number of games wanting amongft them is conlidcrable ; the Operation muft be tedious ; and that there may be fome danger of miftake, in feparating and collecting the fe- veral parts of their Expedations, from the Terms of the Multi- nomi-il : I invented this other Solution, which was publifhed in the Vir*' Book of my Mifcellanea Analytica, A. D. 1730. The Skill of the Gamefters A, B, C, &c. is now fuppofed to be as a, b. c, 8cc. refpedively : and the Games they want of the Set are^, o, r, tec. Then in order to find the Chance of a particular Gamefter, as of A, or his Right in the Stake i, we may proceed as follows. 1°. Write down Unity. 2°. Write down in order all the Letters b, c, d, &c. which denote the Skill of the Gamefters, excepting only the Letter which belongs to the Gamefter whofe Chance you are computing; as in our Exam- ple, the Letter a is omitted. 3°. Combine the fame Letters b, c, d, Sec. by two's, three's, four's, &c. 4°, Of thefe Combinations, leave out or cancel all fuch as make any Gamefter befides A, the winner of the Set ; that is, which give to B, q Games ; to C, r Games, to D, s Games, &;c. r°. Multiply the whole by aP-K 6°. Prefix to each Produdt the Number of its Fernmtatiotis, that is, of the different ways in which its Letters can be written *. • Of Comlinatkns and Pe/muialmi, See Prob. xiv. &/cfj- 7°. Let 7^^ Doctrine (?/* Chances. 51 7°. Let all the Produdls that arc of the fame dimenfion, that is, which contain the fame number of Letters, be coUefted into different fums. 8°. Let thefe feveral Sums, from the lowefl: dimenfion upwards, be divided by the Terms of this Series, //>-', /?, />+■, /i^+2, &c. refpedively: in which Series f='' + ^ + t+'i+ &c. 9°. Laftly, multiply the Sum of the Quotients by ~ , and the Produft {hall be the Chance cr Expedlation required ; namely the Right of y^ in the Stake i. And in the fame way, the Expeifla- tions of the other Gamefters may be computed. Ex AxM PL E. Suppofing /> = 2, $' = 3, ^ = 5 j write, as direded in the Rule, 1, l> -\- c, hb\-bc-\-cc, Ibcc + bc^ + c*, bbcf- -\- bc\ bbc^. Multiply eacii term by «/*—', which in our Example is a-—^, or a; prefix to each Produft the number of its Permutations, dividing at the fame time the fimilar Sums by//—', />, //*+', &c. that is hy f, pipy &c; And the whole multiplied into 4 will give the Ex- pecration ot A^=. — into -r -| j-^ \ j—^ -\- \2abbc-\-\7obcc-\-iaci ■ ',r flf>cc-\-zoabc'-\-^ac* . f<^ahhc'-\--irnbr^ . T^rnh'^r'- Ji -I Ji I -p I —J. • If we now fubftitute for a, b, r, any numbers at pleafure, we fhall have the anfwer that belongs to thofe iuppofed degrees of Skill. As if we make « = i, br= i, f ^= i ; the Expedtation of A will be, L X - A-- 4--A^4--il-4--ii-4--^4- -^^ = -^^-i 3 3 ' 9 ' 27 ' Si ' 245 ' 729 ' 2187 zi6y And, by like Operations, thofe of B and C will be — ^^ and — ^ ' •' ^ ' 21S7 2167 refpedively. PROBLEM Vn. Two Gajnejlers A and B, each having 1 2 Counters^ 'play with three Dice^ on condition that if 1 1 Poijits come up^ B jljqll give one Counter to K\ if i^ Points come up, A fhall give one Counter to B ; ajid that he pall be the winner who pall foonefl get all the Cotmters of his Ad- verfary : what is the Probability that each of them has of winning P H 2 Sol- 5 2 T'he Doctrine (j/" Chances. Solution. Let tlie number of Counters which each of them hasbe^=/i; and let a and b he the number of Chances they have refpedively for getting a Counter, each cafi; of the Dice : which being fuppofed, I fay that the Probabilities of winning are refpeilively as al" to bf \ now becaufe in this cafe/'= 12, and that, by the preceding Lemma, c= 27, and /^r=r 15, it follows that the Probabilities of winning are refpedively as 27'^ to i j'^*, or as 9'* to 5", or as 2H2429536481 to 244140625 : which is the proportion afligned by Huygens in this particular cafe, but without any Demondration. Or more generally : Let p be the number of the Counters of A, and q the number of the Counters of B ; and let the proportion of the Chances be as a to b. I fav that the Probabilities of winning will be refpedively as ai x iif — he to bf X a'' — b"! ; and confequently the Probabi lities them- (ii X o^ b^i , b^ y. ai — bi fclves will be ■ ,, 7—7-, = i?, and — -r- r-v- = S. Demonstration. Let it be fuppofed that A has the Counters E, F, G, H, &c. whofe number is p, and that B has the Counters I, K, L, See. whofe num- ber is q : moreover, let it be fuppofed that the Counters are the thing played for, and that the value of each Counter is to the value of the following as a to b, m fuch manner as that E, F, G, H, I, K, L be in geometric Progreffion ; this being fuppofed, A and B in every circumfl:ance of their Play may lay down two fuch Counters as may be proportional to the number of Chances each has to get a fingle Counter ; for in the beginning of the Play, A may lay down the Counter H, which is the loweft of his Counters, and B the Counter I, which is his higheft ; but H, 1 :: a, b, therefore A and B play upon equal terms. If A beats B, then A may lay down the Counter I which he has juft got of his adverfary, and B the Coun- ter K ; but I, K : : a, b, therefore A and B ftill play upon equal terms. But if A lofe the firfl: time, then A may lay down the Counter G, and B the Counter H, which he juft now got of his adver- fary; but G, H : : a, b, and therefore they ftill play upon equal terms as before : So that, as long as they play together, they play- without advantage or difadvantage. Now the Value of the Expec- tation which A has of getting all the Counters of B is the produdl o£ Hoe Doctrine (?/'Chances. 53 of the Sum he expedis to win, and of the probability of obtaining it, and the fame holds alfo in refpedl to B: but the Expedations of A and E are fuppofed equal, and therefore the Probabilities which they have refpedively of winning, are reciprocally proportional to the Sums they expeft to win, that is, are diredlly proportional to the Sums they are pofTefled of. Whence the Probability which A has of winning all the Counters of B, is to the Probability which B has of winning all the Counters of A, as the Sum of the terms, E, F, G, H, whofe number is/", t o the Su m of the terms I, K, L, whofe number is q, that is as ai x cf — b^ to b^ x a'' — b' ■, as will eafily appear if thofe terms, which are in geometric Progreflion, are actually fummed up by the known Methods. Now the Probabili- ties of winning are not influenced by the Suppofition here made of each Counter being to the following in the proportion of ^ to ^ > and therefore when thofe Counters are fuppofed of equal value, or rather of no value, but ferving only to mark the number of Stakes won or loft on either fide, the Probabilities of winning will be the fame as we have affigned. 'Corollary r. If we fuppofe both a and b'mz ratio of equality, the expreffions whereby the Probabilities of winning are determined will be reduced to the proportion of ^ to y ; which will eafily appear if thofe ex- preffions be both divided by a — b. Corollary 2. If A and B play together for a Guinea a Game, and A has but one fingle Guinea to lofe, but B any number, let it be never fo' large; \i A in each Game has the Chance of 2 to 1, he is more likely to win all the Stock of B than to lofe his fingle Guinea ; and jufl as likely, if the Stock of B were infinite Remark. If /> and g, or either of them be large numbers, it will be con- venient to work by Logarithms. Thus, if A and B play a Guinea a Stake, and the number of Chances which A has to win each fingle Stake be 43, but the num- ber of Chances, which B has to win it, be 40, and they oblige them- felves to play till fuch time as 10 o Stakes arc won or loft ; (the num- ber^ being =: y nr 100, and therefore the Ratio fought being -^|'^°.): Fromi £4- ^^ Doctrine 0/ Chances. From the logarithm of 43 -^^ 1.6334685 Subtract the logarithm of 40 = 1.6020600 Difference = 0.03 14085 Multiply this Difference by the number of Stakes to be played off, 1)1%. 100, the produdt will be 3.1408500, to which anfwers in the Table of Logarithms 1383 ; therefore the Odds that A beats B are 1383 to I. Now in all circumftances wherein yf and B venture an equal Sum, the Sum of the numbers expreffing the Odds, is to their difference, as the Money played for, is to the Gain of the one, and the Lofs of the other. Wherefore, multiplying 1382 difference of the numbers expreffing the Odds by 100, which is the Sum ventured by each Man, and di- viding the produdt by 1384, Sum of the Numbers expreffing the Odds, the Quotient will be, within a trifle, 99 Guineas, and 2 Shil- lings, fuppofing Guineas at 21 /■• If iniltjJ of fuppofing the proportion of the Chances whereby A and B may refpedlively win a Stake to be as 43 to 40, we fuppofe them as 44 to 40, or as 1 1 to 10, the Expedation oi A will be worth above 99 Guineas, 20 Shillings and i Penny. PROBLEM VIII. T^'o Gamejlers A and V>lay by 24 Counters., and play with three Dice., on this conditiofi ; that if 11 Points come up., A jhall take one Counter out of the heap ; z^ 1 4, B jhall take out one \ and he jhall be reputed the wifiJier nvho JImH fconejl get 1 2 Counters. This Problem differs from the preceding in this, that the Play will be at an end in 23 Calls of the Die at moft ; (that is, of thofe Cafts which are favourable either to A or B) whereas in the prece- ding cafe the Counters paffing continually from one hand to the other, it will often happen that y^and B will be in fome of the fame circum- f! ances they were in before, which will make the length of the Play unlimited. SOLUT ION. Taking a and b in the proportion of the Chances which there are ■ to throw 1 1, and 14, let ^ -|" '^ ^^ railed to the 23' Power, that is 1 to I'he Doctrine (?/'Chances. 55 to fuch Power as is denoted by the number of all the Counters want- ing one: then fhall the 12 firfi: terms of that Power be to the 12 laft in the fame proportion as are the Probabilities of winning. PROBLEM IX. Suppofl?ig A and B, 'whofe proportion ojjkill is as a to b, to play together^ till A either wins the number q of Stakes^ or lofes the number p of thern ; and that B fets at every Game the Stun G to the Sum L ; it is required to Jind the Advantage or DiJ advantage of A. Sol ution. Firft, Let the number of Stakes to be won or lofl: on either fide be equal, and let that number be p ; let there be alfo an equality of fkill between the Gameflers : then I fay that the Gain of A will be pp X — \ — • , that is the fquarc of the number of Stakes which either Gamefter is to win or lofe, mukiplied by one half of the diffe- rence of the value of the Stakes. Thus if A and B play till fuch time as ten Stakes are won or loft, and B fets one and twenty Shil- lings to 20; then the Gain of y^ will be 100 times the half difference between 21 and 20 Shillings, viz. 50/"- Secondly, Let the number of Stakes be unequal, fo that A be obliged either to win the number q of Stakes, or to lofe the number p J let there be alfo an equality of Chance between A and B : then I fay that the Gain of A will be pq x — - — ; that is the Produdt of the two numbers of Stakes, and one half the difference of the value of the Stakes multiplied together. Thus if A and B play together till fuch time as either A wins eight Stakes or lofes twelve, then the Gain of yl will be the produft of the two numbers 8 and 12, and of 6 ''^- half the difference of the Stakes, which produdl makes Ihirdly, Let the number of Stakes be equal, but let the number of Chances to win a Game, or the Skill of the Gamefters be unequal, in the p roportion oi a io b ; then I fay that the Gain of A will be Fourthly, Let the numbei* of Stakes be unequal, and let alfo ther number of Chances be unequal : then I fay that the Gain of A will J.T 5 6 I'hQ Doctrine o/" Chances. Demonstration. Let R p.ni S refpcdtively reprefent the Probabilities which A and iB have of winning all tlie Stakes of their Adverfary ; which Pro- babilities have been determined in the vii''' Problem. Let us firffc fuppofe that the Sums depofited by A and B are equal, viz. G, and G: now fince A is either to win the Sum ^G, or lofe the Sum pG, it is plain that the Gain of A ought to be eflimated by R^G — S/)G ; moreover fince the Sums depofited are G and G, and that the proportion of the Chances to win one Game is as a to b, it fol- lows that the Gain of A for each individual Game is - ""J , ^ ' ; and for the fame reafon the Gain of each individual Game would be " ^'L^^ , if the Sums depofited by A and B were refpedlively L and G. Let us therefore now fuppofe that'they are L and G; then in order to find the whole Gain of A in this fecond circum- ftance, we may confider that whether A and B lay down equal Stakes or unequal Stakes, the Probabilities wliich either of them has of winning all the Stakes of the other, fuftcr not thereby any alte- ration, and that the Play will continue of the fame length in both circumftances before it is determined in favour of either ; wherefore the Gain of each individual Game in the firft: cafe, is to the Gain of each individual Game in the fecond, as the whole Gain of the firft cafe, to the whole Gain of the fecond ; and confequently the whole Gain of the fecond cafe will be R^ — S^ x " '~'^ - , or reftoring the values of R and S, ^^ ^ " ~i. ~ \j^'^ " ~ — mul- tiphed by -^:r^ . PROBLEM X. Tl^ree Perjons A, B, C, oui of a heap of 12 Counters^ whereof 4 a?'e white^ and 8 black.^ draw bliftdfold o?je Counter at a ttme^ i?t this manner ; A begins to draw ; B follows A ; C follows B ; then A begins again ; and they continue to draw in the fanie order^ till one of them who is to be reputed the winner .^ draws the frfl white. What are the refpeSiive Probabilities of their '■^inning f SOLU- 7^^ Doctrine o/" Chances. 57 Solution. Let n be the number of all the Counters, a the number of white, b the number of black, and i the whole Stake or the Sum plaved for. 1°. Since A has a Chances for a white Counter, and b Chances for a black, it follows that the Probability of his winning is — ^ = — ; therefore the Expedation he has upon the Stake i, arifing from the circumftance he is in, when he begins to draw, is — X I = — : let it therefore be agreed among the Adventurers, that A (hall have no Chance for a white Counter, but that he fhall be reputed to have had a black one, which fliall adually be taken out of the heap, and that he fhall have the Sum — paid him out of the Stake, for an Equivalent. Now - being taken out of the Stake there will remain i — — = -^^^ = — . n rt n 2°. Since B has a Chances for a white Counter, and that the num- ber of remaining Counters is « — 1, his Probability of winning will be —^ ; whence his Expedation upon the remaining Stake - , arifing from the circumftance he is now in, will be n . n — I Suppofe it therefore agreed that B fhall have the Sum - paid him out of the Stake, and that a black Counter fhall alfo be taken out of the heap. This being done, the remaining Stake will be - — r— — - or " " ' , but 7ib — ab =: bb i wherefore the re- maining Stake is ■ ~' . 3°. Since C has a Chances for a white Counter, and that the number of remaining Counters is n — 2, his Probability of winning will be -^— ^ , and therefore his Expedation upon the remaining Stake arifing from the circumflance he is now in, will be LI • g.i,— ~. „_^; > which we will likewife fuppofe to be paid him out of the Stake, flill fuppofing a black Counter taken out of the heap. 4°. A may have out of the remainder the Sum ' ~' ~'"" n . n — l.n — 2.n— 3 ' and fo of the refl till the whole Stake be exhaufled. I Where- 5-8 ItJoe Doctrine (t/* Chances. Wherefore having written the following general Series, viz. - -\- -^P -1- -^^Q + ^^^R + ^^=^S, &c. wherein P, Q, R, S, 6cc. denote the preceding Terms, take as many Terms of this Series as there are Units in /^ -|- i, (for fince b reprefents the number of black Counters, the number of drawings cannot exceed b ■\- \^ then take for A the firft, fourth, feventh, Sec. Terms ; take for B the fecond, fifth, eighth, &c. for C the third, fixth, &c, and the Sums of thofe Terms will be the refpedtive Expectations of A, B, C ; or becaufe the Stake is fixed, thefe Sums will be proportional to the refpedive Probabilities of winning. Now to apply this to the prefent cafe, make ;; r= 12, ^ = 4, 3 = 8, and the general Series will become—— -1 P -J — Q 4- f R + Is + fT -1- -f U +- ^X ^- -Y : or multiplying the whole by 495 to take away the fradlions, the Series will be 165 -\- . 1204- 84 A- 564- 35 4-20 -V- 10 + 4+ '• Therefore affigning to A 165 -J- 56 -\- 10 = 23 i, to 5 120 -\- 35 -j- 4 = i59i to C 84 4- 20 4- I = 105, the Probabilities of winning will be proportional to the numbers 231, 159, 105, or 77, If there be never fo many Gamefters A, B, C, D, &c. whether they take every one of them one Counter or more j or whether the fame or a different number of Counters; the Probabilities of winning will be determined by the fame general Series. Remark I. The preceding Series may in any particular cafe be (hortened •, for iftf = i, then the Series will be ^ x 14-1 + 14-^-1- ^ + 1 + ^. &c. Hence it may be obferved, that if the whole number of Coun- ters be exadly divifible by the number of Perfons concerned in the Play, and that there be but one fingle white Counter in the whole, there will be no advantage or difadvantagc to any one of them from the fituation he is in, in refped to the order of draw- ing- If^=2, then the Series will be — — x «— i-j-«— 2-h«— 3-1-7;— 4+«— 5, &c. If ^ = 3, then the Series will be „ . „_^i . „_^ x « — I . n — z-\-n — 2 . n — 3 4~'^—' 3 • '^"-4> ^c. If 7he Doctrine <?/" Chances. If = 4, then the Series will be -;; — T^i — \ 59 -2 . « - 3 &C. the feveral X » — I . « — 2 . n — 3 -4- n — 2 . « — 3 . n — 4, Wherefore rejeding the common Multiplicators ; Terms of thefe Series taken in due order, will be proportional to the feveral Expectations of any number of Gamefters : thus in the cafe of this Problem where « = 12, and ^ = 4, the Terms of the Series will be. For A 1 1 X 10 X 9 = 990 8 X 7x6 = 336 5 X 4 X 3= ^o ,386 For B 10 X 9x8 — 720 7 X 6x5 210 4 X 3 x.s — 24 9 X 6 X 1 For C 8x7 = 504 5 X 4 = 120 X 2 X I = 6 954 630 Hence it follows that the Probabilities of winning will be refpedive- ly as 1386, 954, 630, or dividing all by 18, as 'j']^ 53, 35, as had been before determined. Remark 2. But if the Terms of the Series are many, it will be convenient to fum them up by means of the following Method, which is an imme- diate confequence of the fifth Lemma of Sir Ijaac Neivtons Princi- pia. Book 111 j and of which the Demonftration may be deduced from his Analyfts. If there be a Series of Terms, A, B, C, D, E, &c. let each Term be fubtradled from that which immediately follows it, and let the Remainders be called firft Differences, then fubtrad each difference from that which immediately follows it, and let the re- mainders be called fecond differences ; again, let each fecond diffe- rence be fubtraded from that which immediately follows it, and let the remainders be called third differences, and fo on. Let the fiifl: of the firil Difference? be called d, the firft of the fecond </, the firfl: of the third d, Sec. and let x be the interval between the firft Term A, and any other Term, fuch as E, that is, let the num- ber of Terms from A to E, both inclufive, be x -|- j , then the Term k-Vxd-\- I-x^^-f .V X r — I X v ■ I . 2 . ■(/, Sec. From hence it manifeftly follows, that Itt the number of Terms between A and E I 2 be 6o The Doctrine o/" Chances. be never fo great, if it fo happen that all the differences of one of the orders are equal to one another, the following differences of all the other orders will all be = o ; and that therefore the lall: Term will be affignable by fo many Terms only of the Series above-writ- ten, as are denoted by the firft Difference that happens to be = o. This being premifed, it will be eafy to fliew, how the Sums of thofe Terms may be taken ; for if we imagine a new Scries whereof the firft Term ftiall be = o ; the fecond =r A ; the third = A -^ B ; the fourth = A + B -|- C ; the fifth = A -f B + C + D, and fo on ; it is plain that the affigning one Term of the new Series is find- ing the Sum of all the Terms A, B, C, D, &c. Now fince thofc Terms are the differences of the Sums o. A, A-)- B, A-|- B -|- C, A-^- B -1- C -|- D, &c. and that by Hypothefis fome of the differences of A, B, C, D, are = o, it follows that fome of the differences of the Sums will alfo be r= o j and that whereas in the Series K-\- xd ■\- -J, &c. whereby a Term was afligned, A reprefented * X — X I 2 the firft Term, d the firft of the firft differences, d the firft of the fe- cond differences, and that x reprefented the Interval between the firft Term and the laft, we are now to write o inftead of A ; A in- ftead oi d', d inftead oi d ; d inftead of ^, &c. and x + i inftead of X ; which being done the Series expreffing the Sums will be * -f- I . * i I AT + 1 . AT ■ ;y — 1 o -}- ^ + 1 X A -V -~~;i + i ~r. 3 dy &c. or X -|- 1 X A 4- i^-f . ""-' d + "•"-••""' ;/, &c. where it will not '2 '2.3 '2.3.4 perhaps be improper to take notice, that the Series by me exhibited in my firft Edition, though feemingly differing from this, is the fame at bottom. But to apply this to a particular cafe, let us fuppofe that three Perfons yf, B, C playing on the fame conditions as are expreffed in this x'*" Problem, the whole number of Counters were 100, inftead of 12, ftill preferving the fame number 4 of white Counters, and that it were required to determine the Expedlations of j4, B, C. It is plain from what has been faid in the firft Remark, that the Expedtation of A will be proportional to the fum of the num- bers 99 X 98 X 97 4- 96 X 95 X 94 + 93 ^ 92 X 91 -f 90 X 89 X 88, &c. * that The Doctrine ^Chances. 6i that the Expedation of B will be proportional to the Sum of the nua.b;rs 98 X 97 X 96 -j- 95 X 94 X 93 -|- 92 X 91 X 90 + 89 >*^ S8 X 87, &c. and laftly, that the Expedation of C will be proportional to the Sum of the numbers 97x96 X 95 4- 94 X 93 X 92 4-91 X 90 X 89 -|- 88 X 87 X 86, &c. But as the number of Terms which conftitute thofe three Series is equal to the number of black Counters increaled by i, as it has been obferved before, it follows that the number cf all the Terms diftributed among A^ 4^, C, mufl be 97 ; now dividing 97 by the number of G.imefters which in this cafe is 3, the quotient will be 32 ; and there remaining i after the divifion, it is an indic.tion that 33 Terms enter the Expedtation of -^, that 32 Terms enter the Expedtation of B, and 32 likewife the Expeftation of C; from whence it follows that the laft Term of thofe which belong to A will be 3 X 2 x I, the lad of thoie which belong to B will be 5 x 4 x 3, and the laft of thofe which belong to C will be 4 x 3 x 2. And therefore if we invert the Terms, making that the firft which was the laft, and take the differences, according to what has been pre- fcribed, as follows ^ A 7> X 2 X 1 — 6 6 X 5 X 4 120 9 X 8 X 7 — 504 12 X ji X 10 1320 15 X &c. H X 13 _ 2730 d 114 d "d 3^4- 270 itz 816 M'^ 162 1410 594 then the Expedation of ^, as deduced from the general Theorenr, will be exprefled by X 4- 1 X 6 H ; h -77-T— X 270 + ^ ^ ^ X 162 t which being contraded, then reduced into its fadors, will be equi- valent to -^ X X- -f IX AT -j- 2 X 3x4- I x 3X-J-4. In like manner, it will be found that the Expedation of 5 is equi- valent to And 62 T'he Doctrine o/" Chances. And that the Expedlation of C is equivalent to - X X -)- I X a; -^- 2 X <^xx -\- z'jx -\- 1 6. Now X in each cafe reprefents the number of Terms wanting one, which belong feverally to A^ B, C ; wherefore making x J- 1 = />, the feveral Ex'pedations will now be exprefled by the number of Terms which were originally to be fummed up, and which will be as folUows, For A, p xPjJzJJ^JP ~J:J^ J/lij. For B, p \ p -\- i X 3/» -}- 2 X 2/» -|- 5 For C, /> X /) -{- I X (jpp + gp — ^ But ftill it is to be confidered, that p in the firft cafe anfwers to the number 33, and in the other two cafes to 32 ; and therefore/* being interpreted for the feveral cafes as it ought to be, the feveral Ex- pcdtations will be found proportional to the numbers 41225, 39592, 38008. If the number of all the Counters were 500, and the number of the white ftill 4, then the number of all the Terms icprefenting the E>:pedl.uions of A^ B., C would be 497 : now this number be- ing divided by 3, the Quotient is J65, and ihe remainder 2. From whence it follows that the Expectations of A and B confift of 1 66 terms each, and the Expedlation of C only of 165, and therefore the loweft Term of all, viz. 3x2x1 will btlon^' to B, the laft but one 4x1x2 will belong to A, and the laft but two will belong toC. PROBLEM XI. 7^ A, B, C ihrow in their turns a revular Ball having 4 u'hit^ faces and eight black ones ; and he be to be re- puted the winner ivbo Jhall firjl bring up one of the ivhite faces ; // is demanded^ what the proportion is of their refpcciive Probabilities of 'wi?7nmg f Solution. The Method of reafoning in this Problem is exadly the fame as that wiiich we have made ufc of in the Solution of the preceding : but whereas the different throws of the Ball do not diminifh the number of its Faces j in the room of the quaii titles b — i, b — 2, "the Doctrine f?/" Chances. 63 b — 3, &c. n — I, n — 2, n — 3, &c. employed in the Solution of the aforefaid Problem, we mufl: fubftitute b and n refpedtiveh\ and the Series belonging to that Problem will be changed into the following, which we ought to conceive continued infinitely. a , ab , abb , ab^ , ab* , ab^ g _ _ U — -\ , — , &c. then taking every third Term thereof, the refpedlive Expedatlons of ^, ^, C will be expreffed by the following Series, a , ohT: . ab'i . al') . ah*'- . T ^- ^ -V -;r -\- IT'-- -1--^' &c- ,m r „5 n^ ,fi T^ „.. -r „'4 > '^^' abb , alS . ab^ , a'" . ab'* - But the Terms, whereof each Series is compofed, are in geometric Progreflion, and the ratio of each Term in each Series to the fol- lowing is the fame j wherefore the Sums of thefe Series are in the fame proportion as their firft Terms, i^iz, as — , -^ , ■—- , or as nn, bn, bb ; that is, in the prefent cafe, as 144, 96, 64, or 9, 6, 4. Hence the refpedllve Probabilities of winning will likewife be as the numbers 9, 6, 4. Corollary i. If there be any other number of Gamefters A, B, C, D, &c. playing on the fame conditions as above, take as many Terms in the proportion of n to b, as there are Gamefters, and thofe Terms will refpedlively denote the fcveral Expedtations of the Gamefters. Corollary 2. If there be any number of Gamefters A, 5, C, D, &c. playing on the fame conditions as above, with this difference only, that all the Faces of the Ball fhall be marked with particular figures i, 2, 3, 4, &c. and that a certain number p of thofe Faces ftiall intitle A to be the winner ; and that likewife a certain number of them, as q, r, s, t, &c. fhall refpedively intitle B, C, D, E, Sec. to be winners : make n — p -^ a, n — q '=^ b, ti — rr=.c, n — s =z d, n — t :=. e, 6cc. then in the following Series ; p , f/a , Tab , sabc tabcd - - A^ -^ — , &c. the Terms taken in due order will refpedively reprefent the feveral Probabilities of winning. For 64 Ihz Doctrine (j/* Chances. For if the law of the Play be fuch, that every Man having once played in his turn, (hall begin regularly again in the fame manner, and that continually, till fuch time as one of them wins ; we are to take as many Terms of the Series as there are Gamefters, and thofe Terms will reprefent the refpedive Probabilities of winning. But the Reafonof this Rule will beft appear if we apply it to fome cafy Example. Let therefore the three Gamefters y/, B, C throw a Die of 12 faces in their Turns; of which 5 faces are favourable to y^, 4 faces are favoura- ble to B, and the remaining 3 give the Stake to C. Then^:^:^, g := 4., r =: 3 : and there being but 3 Gamefters, the fame Chances, and in the fame Order yi, B, C, will recur perpetually after a Round of three throws, till the Stake is won ; or rather, as we fuppofe in the demonlhation, till the Stake is totally exhauft.ed, bv each Gamefter, inftead of his thiow, taking out of it the part to which the chance of that throw entitles him. Now A having p Chances out of », or 5 out of 1 2, to get the whole Stake at the firft Throw, let him take out of it the Value of this Chance - : and there will remain i — - = -^^ = - to be thrown K ' ft n n for by B. And B's Chances for winning in his Throw being g out ofn, or 4 out of 12, the Value of his prefent Expedation is -^ X — := -— - ; which if he takes out of the Stake — there will remain — — -^ = — X I — — :== — x-,tobe thrown for by C. n nun' ■> His Chances for getting this Stake being r out of «, or 3 out of 12, the Value of his Expedation is -^ ; which he may take out of the Stake -^ : and refign the Die to ^, who begins the fecond Round. But if, for the Stakes that remain after the firft, fecond, third, &c. Rounds, we write R', R'. R'", &c. rtfpedtively, it is manifeft that the Value of a Gamefter's Chance in each Round is proportional to the Stake R', R", R", &c. which remained at the beginning of that Round. Thus the Value of ^'s firft Throw having been ^ X i, the Value of his fecond will be ^ x R', of his third, -^ x R", &c. And the Value of £'s firft Throw having been -^ X i, that of his fecond will ^ X R', of his third, -^ x R", 6cc. and the like for the feveral Expedations of C. Put 7^^ Doctrine <?/"Chances. 65 Put ^ — I 4- R' -j- R" -f R'", &c. and the Total of As Ex- pectations will be ^x5;of5, -^ X 5; of C, -^ X 5 ; or re- jeding the common Faftor 5, the Expedtations of /^, B, C, at the beginning of the Play will be as -^ , -^ , ^ , refpedively : that is as the 3 firft Terms of the Series. And the like reafoning will hold, be the Number of Gamefters, their favourable Chances, or order of Throwing, what you will. In the prefent Example, t = J- ~ -^ .^ Jl. — -1^ — JlL . SfL — .'J!L : and the Chances of A B, C, refpedively, are 1728 * «' 1728 as the Numerators 720, 336, 168; that is, as 30, 14, 7. or the whole Stake being 5 1 pieces, A can claim 30 of them, B 14, and C the remaining 7. In making up this Stake, the Gamefters A, B, C, were, at equal play, to contribute only in proportion to their Chances of winning ; that is in the proportion of p,q, r, or 5, 4, 3, refpedlively : and, be- fore the Order of throwing was fixt, their Chances mufl have been exadtly worth what they paid in to the Stake. What gives A the great advantage now is, an antecedent good luck of being the firft to throw. If B had been the firft ; or if yl, taking his firft Throw, had mift of a / face, then 5's Chance had been the better of the two. And if it were the Law of Play that every Man {hould play feveral times together, for inftance twice : then taking for yf the two firft Terms, for B the two following, and fo on ; each couple of Terms will reprefent the refpedlive Probabilities of winning, obferving now that p and q are equal, as alfo r and s. But if the Law of Play fliould be irregular, then you are to take for each Man as many Terms of the Series as will anfwer that irre- gularity, and continue the Series till fuch time as it gives a fufficient Approximation. Yet if, at any time, the Law of the Play having been irregular, fhould afterwards recover its regularity, the Probabilities of winning, will (with the help of this Series) be determined by finite expref- fions. Thus if it fhould be the Law of the Play, that two Men A and B having played irregularly for ten times together, tho' in a manner agreed on between them, they fliould alfo agree that after ten throws, they fhould play alternately each in his turn : diftribute the ten firft Terms of the Series between them, according to the order fixed upon by their convention, and having fubtraded the Sum of thofe Terms K from 66 The Doctrine ^Chances. from Unity, divide the remainder of it between them in the propor- tion of the two following Terms, which add refpedively to the Shares they had before ; then the two parts of Unity which A and B have thus obtained, will be proportional to their refpedlive Proba- bilities of winning. PROBLEM XII. There are any number of Gamejiers^ who in their Turns., which are decided by Lots., tur7i a Cube., having 4 of its Faces marked T, P, D, A, the other two Faces which are oppojjte have each a little Kfiob or Pivet, about which the Cube is made to turn ; the Gamejiers each lay dowJi a Sum agreed upon, the firfl begins to turn the Cube ; 7tow if the Face T be brought up, he fweeps all the Mo?iey upon the Board, and the?i the Play begins anew ; if any other Face is brought up, he yields his place to the next Man, but with this difference, that if the Face P comes up, he, the frfl Man, puts down as much Money as there was upon the Board ; if the Face D comes up, he neither takes up aity Money nor lays down ajty ; if the Face A cofms up, he takes up half of the Mofiey up07i the Board ; whe7i every Man has played in his Turn upon the fame co7tditiom as above, there is a recurrency of Order, whereby the Board may be very much enlarged, viz. if it fo happen that the Face T is intermitted during many Trials : now the i^ue- fiion is this ; when a Gafnefler comes to his Turn, fup- pofng him afraid of laying down as much Money as there is already, which may be confderable, how mufi he compound for his Expe&ation with a SpeElator willing to take his place. SoLUT I ON. Let us fuppofe for a little while that the number of Gamefters is infinite, and that what is upon the JBoard is the Sum /j then, there Hoe DOCTRIN E ^ Ch A NCES. 67 there being i Chance in 4 for the Face T to come up, it follows that the Expedation of the firft Man, upon that fcore, is ^ 2°. There being i Chance in 4 for the Face P to come up, whereby he would neceflarlly lofe f^ (by reafon that the number of Game- fters having been fuppofed infinite, his Chance of playing would never return again) it follows that his Lofs upon that account ought to be eftimated by —f. 3°. There being i Chance in 4 for the Face D to come up, whereby he would neither win or lofe any thing, we may proceed to the next Chance. 4°. There being i Chance in 4 for the Face A to come up, which intitles him to take up — / his Expedlation, upon that account, is -^/, or fuppofing 8 = », his Expedtation is — /; now out of the four cafes above- mentioned the firft and fecond do defiroy one another, the third nei- ther contributes to Gain or Lofs, and therefore the clear Gain of the firft Man is upon account of the fourth Cafe ; let it therefore be agreed among the Adventurers, that the firft Man fhall not try his Chance, but that he (hall take the Sum —/"out of the common Stake f, and that he fhall yield his Turn to the next Man. But before I proceed any farther, it is proper to prevent an Objedtion that may be made againft what I have aflerted above, viz. that the Face D happening to come up, the Adventurer in that cafe would lofe nothing, becaufe it might be faid that the number of Gamefters being infinite, he would neceffarily lofe the Stake he has laid down at firft; but the anfwer is eafy, for fince the number of parti- cular Stakes is infinite, and that the Sum of all the Stakes is fuppofed only equal to f, it follows that each particular Stake is nothing in comparifon to the common Stake f, and therefore that common Stake may be looked upon as a prefent made to the Adventurers. Now to proceed ; I fay that the Sum — / having been taken out of the common Stake f, the remaining Stake will be -^^ / or — , fuppofing n — I =d : but by reafon that the firft Man was allowed .— part of the common Stake, fo ought the next Man to be allowed — part of the prefent Stake —f, which will make it that the Expedation of the fecond Man will be — :/; Again, the Expec- K 2 tation 68 Tl^e Doctrine (t/" Chances. tation of the fecond Man being to the Expectation of the firft as — to I, the Expedation of the third muft be to the Expedlation of the fecond alfo as — to i, from whence it follows that the Expec- tation of the third Man will be -—fy and the Expedation of the fourth — /J and fo on ; which may fitly be reprefented by the Series 123456 //„^o -^ J. ^ _|. i!l _|- 4- _|. 4- J_ i!- , &c. Now the Sum of that infinite Series, which is a Geometric Progrefllon, is —Z7-, hut d having been fuppofed -=.11 — i, then n — ^= i, and therefore the Sum of all the Expedations is onlyy^ as it ought to be. Now let us fuppofe that inftead of an infinite number of Game- ftcrs, there are only two ; then, iii this cafe, we may imagine that the firft Man has the firfl, thirds ffthy feventh Terms of that Se- ries, and all thofc other Terms in infinitum which belong to the odd places, and that the fecond Man has all the Terms which be- long to the even places j wherefore the Expectation of the firft Man is ^ m^o I 4- -^ -i- -^ + ^ + — , &c. and the Expecta- tion of the fecond h -^ into i •\- -^ -\- — + ~ + ^r > &c. r df and therefore the Ratio of their Expectations is as -^ to -^ , or as 1 to - , that is as « to « — i, or as 8 to 7 ; and therefore the Expectation of the firft Man is -^/ and the Expectation of the fecond Man is — /; and therefore if a Spectator has a mind to Q take the place of the firft Man, he ought to give him — / But if the number of Gamefters be three, take a third propor- tional to « and d. which will be — , and therefore the three Ex- peCtations will be refpeCtively proportional to «, d^ —^ , or to nn^ dn, dd, and therefore the Expectation of the firft Man is -^ — -~f ' ' ^ tin -i- a/t + aa J which in this cafe is = -7— -A Univerfalfy, Let p be the number of Adventurers, then the Sum for which the Expedation of the firft Man may be transferred to an- The Doctrine o/" Chances. 69 'ithe Game of Bassette. Rules of the Play. The Dealer, otherwife called the Banker, holds a pack of 52 Cards, and having fhuffled them, he turns the whole pack at once, fo as to difcover the laft Card ; after which he lays down by couples all the Cards. The Setter, otherwife called the Tonte^ has 1 3 Cards in his hand, one of every fort, from the King to the Ace, which 13 Cards are called a Book ; out of this Book he takes one Card or more at plea- fure, upon which he lays a Stake. The Ponte may at his choice, either lay down his Stake before the pack is turned, or immediately after it is turned ; or after any number of Couples are drawn. The firft cafe being particular, fhall be calculated by itfelf ; but the other two being comprehended under the fame Rules, we fhall begin with them. Suppofing the Ponte to lay down his Stake after the Pack is turned, I call I, 2, 3, 4, 5, &c. the places of thofe Cards which follow the Card in view, either immediately after the pack is turned, or after any number of couples are drawn. If the Card upon which the Ponte has laid a Stake comes out in any odd place, except the firft, he wins a Stake equal to his own. If the Card upon which the Ponte has laid a Stake comes out in any even place, except the fecond, he lofes his Stake. If the Card of the Ponte comes out in the firft place, he neither wins nor lofes, but takes his own Stake again. If the Card of the Ponte comes out in the fecond place, he does not lofe his whole Stake, but only a part of it, viz. one half, which to make the Calculation more general we fliall call_y. In this cafe the Ponte is faid to be Faced. When the Ponte chufes to come in after any number of Couples are down ; if his Card happens to be but once in the Pack, and is the very laft of all, there is an exception from the general Rule ; for tho' it comes out in an odd place, which fhould intitle him to win a Stake equal to his own, yet he neither wins nor lofes from that circum- ftance, but takes back his own Stake. PRO- 7© 7^^ Doctrine o/" Chances. PROBLEM XIII. 'To ejlwtate at Baflette the Lofs of the Ponte laider any circiimjlance of Cards retnaining in the Stocky when he lays his Stake \ a?id of afiy number of times that his Card is repeated in the Stock. The Solution of this Problem containing four cafes, viz. of the Ponte's Card being once, twice, three or four times in the Stock; we fliall give the Solution of all thefe cafes feverally. Solution of the firjl Cafe. The Ponte has the following chances to win or lofe, according to the place his Card is in. Chance for winning o Chance for lofing y Chance for winning i Chance for lofing i Chance for winning i Chance for lofing i Chance for winning o It appears by this Scheme, that he has as many Chances to win i as to lofe I, and that there are two Chances for neither winning or lofing, viz. the firft and the laft, and therefore that his only Lofs is upon account of his being Faced: from which it is plain that the num- ber of Cards covered by that which is in view being called n, his Lofs will be ^ , or ~ , fuppofing ;' = 7 . Solution of the fecond Cafe. By the firft Remark belonging to the x'*" Problem, it appears -f- that the Chances which the Ponte has to win or lofe are proportional to the numbers, n — i, n — 2, n — 3, &c. Wherefore his Chances for win- ning and lofing maybe expreued by the following Scheme. f Namely, by calling the Ponte's two Cards two white Counters, drawn for alter- nately by A and B ; and fuppofing all A's Chances to belong to the Banker's right hand, and thofe of B to his left. And the like for the Cafes of the Ponte's Card being in the Stock 3 or 4 times, I 1 I 2 I 3 I 4 1 5 I 6 I ^ I Tie Doctrine o/'Chances. 71 I 2 3 4 5 6 7 8 9 * « — I Chances for winning o n — 2 Chances for lofing y n — 3 Chances for winning i n — 4 Chances for lofing 1 n — 5 Chances for winning i « — 6 Chances for lofing i n — 7 Chances for winning i n — 8 Chances for lofing i n — 9 Chances for winning i I Chance for lofing i Now fetting afide the firft and fecond number of Chances, it will be found that the difference between the 3'' and 4''' is = i, that the difference between the 5'*" and 6''' is alfo = i, and that the difference between the 7"* and S'*" is alfo = i, and fo on. But the number of differences is "~^ , and the Sum of all the Chances is — X "~^ : wherefore the Gain of the Ponte is — ^^^-r- • But 12 fix n — I I X 2 his Lofs upon account of the Face is n — 2 x y divided by that is ^''~'^^y : hence it is to be concluded that his Lofs upon the » X « — I * whole is^-l^niiSti^iii or — I — fuppofina; y = — . nxn — I B X ffl — I ^^ ^ •' 2 That the number of differences is -^-^^ will be made evident from; 2 two confiderations. Firft, the Series « — 3, n — 4, n — 5, &c. decreafes in Arith- metic Progrefllon, the difference of its terms being Unity, and the laft Term alfo Unity, therefore the number of its Terms is equal to the firft Term n — 3 : but the number of differences is one half of the number of Terms ; therefore the number of differences is " — % 2 Secondly, it appears, by the x'*" Problem, that the number of all the Terms including the two firft is always 6 ~{- i, but a in this cafe is = 2, therefore the number of all the Terms \s n — i ; from which excluding the two firft, the number of remaining Terms will be K — 3, and confequently the number of differences "~ . That the Sum of all the Terras is - x ""' , is evident alfo' I 2 ' from two different confiderations. Firft 7 2 'The Doctrine (p/" Chances. Firft in any Arithmetic Progreflion whereof the firfl: Term is n — I, the difference Unity, and the laft Term alfo Unity, the Sum of the Progreflion will be — x "~ ' . Secondly, the Series 7-^ x « — 1 + « — 2 -|- /i — 3, &c. mentioned in the firfl: Remark upon the tenth Problem, exprefles the Sum of the Probabilities of winning which belong to the feveral Game- fters in the cafe of two white Counters, when the number of all the Counters is n. It therefore expreffes likewife the Sum of the Proba- bilities of winning which belong to the Ponte and Banker in the pre- fent cafe : but this Sum muft always be equal to Unity, it being a cer- tainty that the Ponte or Banker muft win ; fuppofing therefore that n — \ A^n — 2 -|- « — 3, &c. is = S, we fhall have the Equation 2S •= I, and therefore S = 7 x — ^ — . Solution of the third Cafe. By the firft Remark of the tenth Problem, it appears that the Chances which the Ponte has to win and lofe, may be exprefled by the following Scheme. I 2 4 5 6 7 8 n — t X «— 2 for winning o n — 2 X n — 3 for lofing v n — 3 X « — 4 for winning i n — 4 X n — 5 for lofing i n — 5 X n — 6 for winning i «— 6 X « — 7 for lofing i ;/ — 7 X n — 8 for winning i n — 8 X n — 9 for lofing i 2 X I for winning i '5 Setting afide the firft, fecond, and laft number of Chances, it will be found that the difference between the 3'' and 4"' is zn — 8 ; the difference between the 5'*" and b^*", zn — 12 ; the difference be- tween the 7'*" and 8 *", 2«— 16, &c. Now thefe differences con- ftitute an Arithmetic Progreflion, whereof the firft Term is zn — 8, the common difference 4, and the laft Term 6, being the difference between 4x3 and 3x2. Wherefore the Sum of this ProgrelTion to which adding the laft Term 2x1, which is is «— t; favourable to the Ponte, the Sum total will be — — ^ x ^ — - : but the 7^5 Doctrine <?/ Chances. 73 the Sum of all the Chances is 7 x -^^ x -^^^ , as may be col- leded from the firft Remark of the x'*' Problem, and the laft Pa- ragraph of the fecond cafe of this Problem: therefore the Gain of the Ponte is ? ■ » — 3 • "~^- . But his Lofs upon account of the rv.-, :<5 .^ .n — z.n—^^y or ^y-"— 3 therefore his Lofs upon the whole is -„,„_, — z.n .n-\ . n- fuppofing jy = — l y • " — 3 3 ■ "—3 ■ '•- 3 . 3"— 9 z.n. a— I . »— 2 Solution of the fourth Cafe. The Chances of the Ponte may be expreffed by the following Scheme. I 2 3 4 5 6 7 * ?2 — I X 72 — 2 X n — 3 for winning o n—Zyt.n — 3x« — 4forlofing y n—2 X «— 4 X «— 5 for winning i n — 4 X « — '5 X « — 6 for lofing i « — 5 X ?2 — 6 X « — 7 for winning i n — 6 X n — 7 X « — 8 for lofing i n — 7 X n — 8 X n — 9 for winning i 3 X 2 X I for lofing i Setting afide the firfl and fecond numbers of Chances, and tak- ing the diff rences between the 3'' and 4''', 5'^ and 6''', f^ and 8''', the lall of thefe differences will be found to be 1 8. Now if the number of thofe differences be p, and we begin from the lafl 1 8, their Sum, from the fecond Remark of the x''' Problem, will be found to be /» X /) -|- I X 4/> -|- 5, but p in this cafe is = \\ and therefore the Sum of thefe differences will eafily appear to be -^;=^ X -^=^ X -^^^-1- , but the Sum of all the Chances is 2 2 : ' — X "~' X '*""" X "" -- ; wherefore the Gain of the Ponte is ~ — ? • " — 3 • 2"— ; . j^Q^ j^ig LQ^g upon account of the Face is n n — i.n — 2.« — 3 f "~V '"^ ' ""•* • '^ , and therefore his Lofs upon the whole will be " — 4 ■ 4v n — 5 . 2« — 5 %n—q f f 1 T-^nr; ir:itnrzT ''' — --IT^f^T' fuppofingjy=- . L There 74 ^^ Doctrin'e of Chauces, There ftill remains one fingle cafe to be confidered, viz. what the Lofs of the Ponte is, when he lays a Stake before the Pack is turned up : but there will be no difficulty in it, after what we have faid ; the difference between this cafe and the reft being only, that he is liable to be faced by the firft Card difcovered, which will make his Lofs to be — ~^ ;^— , that is, interpreting ?i by the num- ber of all the Cards in the Pack, viz. 52, about -—-- part of his Stake. From what has been faid, a Table mayeafily be compofed, {hew- ing the feveral Lofles of the Ponte in whatever circumftance he may happen to be. • ATa- the DOCTR A Table INE <?/" Chances, for Bassette. N 52 51 49 47 45 43 41 39 37 35 33 31 29 27 25 23 21 ^9 17 15 13 II 9 7 I 2 3 4 866 867 801 * * * * * * * * * 1735 * * * * * * 98 2352 1602 1474 94 2162 717 (>75 90 1980 1806 ^351 1234 86 617 561 82 1640 1122 1015 78 1482 507 74 1332 914 457 70 1190 818 409 363 321 66 1056 727 ei 930 642 58 54 812 702 562 281 487 243 50 46 600 418 209 506 420 354 295 ^77 42 H7 38 342 272 242 194 121 34 30 97 210 151 75 26 22 156 114 82 56 35 57 110 41 28 '7 18 72 14 42 The 76 The Doctrine o/" Chances. The ufe of this Table will be bed explained by fome Examples. Example i. het it be propofed to find the Lojs of the Ponte, ivhen there are 26 Cards remaining in the Stock, and his Card is twice in it. In the Column N find the number 25, which is lefs by i than the number of Cards remaining in the Stock : over-againft it, and under the number 2, which is at the head of the fecond Column, you will find 600 ; which is the Denominator of a fradion whofe Numerator is Unity, and which fliews that his Lofs in that circum- flance is one part in fix hundred of his Stake. Example 2. To find the Lofs of the Ponte when there are eight Cards remaining in the Stock, and his Card is three times in it. In the Column N find the number 7, lefs by one than the num- ber of Cards remaining ia the Stock : over-againft 7, and under the number 3, written on the top of one of the Columns, you will find 35, which denotes that his Lofs is one part in thirty-five of his Stake. COROL L ARY I. 'Tis plain from the conflrudlion of the Table, that the fewer Cards are in the Stock, the greater is the Lofs of the Ponte. Corollary 2. The leaft Lofs of the Ponte, under the fame circumftances of Cards remaining in the Stock, is when his Card is but twice in it j the next greater but three times ; ftill greater when four times ; and the greateft when but once. If the Lofs upon the Face were varied, 'tis plain that in all the like circumftances, the Lofs of the Ponte would vary accordingly ; but it would be eafy to compofe other Tables to anfwer that variation ; fince the quantity y, which has been afiiimed to rcprefent that Lofs, having been preferved in the general exprefiion of the Lofi^es, if it be interpreted by — inftead of — , the Lofs, in that cafe, would be as eafily determined as in the other: thus fuppofing that 8 Cards are remaining in the Stock, and that the Card of the Ponte is twice in it, and alfo that y fliould be interpreted i The Doctrine c/'Chances. 77 interpreted by — , the Lofs of the Pontc would be found to be —- inftead of -^ . 63 42 The Game of Pharaon. The Calculation for Pharaon is much like the preceding, the rea- fonings about it being the fame ; it will therefore be fufHcient to lay down the Rules of the Play, and the Scheme of Calculation. Rules of the Play. Firft, the Banker holds a Pack of 52 Cards. Secondly, he draws the Cards one after the other, and lays them down at his right and left-hand alternately. Thirdly, the Ponte may at his choice fet one or more Stakes upon one or more Cards, either before the Banker has begun to draw the Cards, or after he has drawn any number of couples. Fourthly, the Banker wins the Stake of the Ponte, when the Card of the Ponte comes out in an odd place on his right-hand ; but lofes as much to the Ponte when it comes out in an even place on his left-hand. Fifthly, the Banker wins half the Ponte's Stake, when it happens^ to be twice in one couple. Sixthly, when the Card of the Ponte being but once in the Stock, happens to be the laft, the Ponte neither wins nor lofes. Seventhly, the Card of the Ponte being bui twice in. the Stock, and the lafl couple containing his Card twice, he then lofes his whole Stake. PROBLEM XIV. To Ji7td at Pharaon thz Gain of the Banker in any cir- cumflance of Cards remaining in the Stock, and of the nmnber of times that the Ponte s Cards is contained in it. This Problem having four Cafes, that is, when the Ponte's Card is once, twice, three, or four times in the Stock j we (hall give the Solution of theie four cafes feverally. S aL tt- 78 T'hQ Doctrine i?/* Chances. Solution of the jirjl Cafe. The Banker has the following number of Chances for winning and lofing. Chance for winning i Chance for lofing i Chance fur winning i Chance for lofing i Chance for winning i Chance for lofing o Wherefore, the Gain of the Banker is — , fuppofing n to be the number of Cards in the Stock. Solution of the fecond Cafe. The Banker has the following Chances for winning and lofing. I 2 3 4 5 * I 2 3 4 5 6 7 8 * \ n-^ 2 Chances for I Chance for n — 2 Chances for { n — 4 Chances for I Chance for }i — 4 Chances for winning i winning y lofing I winning i winning y lofing I r 6 Chances for I Chance for 6 Chances for winning i winning y lofing I {" — 8 Chances for I Chance for n — 8 Chances for winning i winning y lofing I 1 Chance for winning i The Gain of the Banker is therefore n — 2 . V + or — n-l" I ;i^;^^ fuppofing _)rr=i.. The only thing that deferves to be explained here, is this ; how It comes to pafs, that whereas at Baffette, the firft number of Chances for winning was reprefented by n — i, here 'tis reprefented by n — 2 > to anfwer this, it mufl be remembered, that according to the Law of The Doctrine (P^Chances. 79 of this Play, if the Ponte's Cards come out in an odd place, the Banker is not thereby entitled to the Ponte's whole Stake : for if it fo happens that his Card comes out again immediately after, the Banker wins but one half of it ; therefore the number n — i is di- vided into two parts, 11 — =• 2 and i , whereof the firft is proportional to the Probability which the Banker has for winning the whole Stake of the Ponte, and the fecond is proportional to the Probability of winning the half of it. Solution of the third Cafe. The number of Chances which the Banker has for winning and lofing, are as follow : I 2 3 4 5 6 7 {" 2 X n — 3 Chances 2 X n — 2 Chances 2 X n — 3 Chances for wlnnins' i for winning y for lofing I 1 n 4 X n — 5 Chances 2 X n — 4 Chances 4 X n — 5 Chances 1" 6 X 71 — 7 Chances 2 X n—t Chances 6 X 11 — 7 Chances tor winning i for winning y for lofing I for winning i for winning y for lofing I i 8 '^ n — 9 Chances 2 ^ «— 8 Chances 2 X I Chances for winning i for winning y for lofing I Wherefore the Gc.in of the Banker is , or "- fuppofing y^=-^. The numl of Chances for the Banker to win, is divided into two parts, whereof the firft exprefles the number of Chances he has for winning the whole Stake of the Ponte, and the fecond for winning the half of it. Now for determining exadliy thofe two parts, it is to be con- fidered, that in the firft couple of Cards that are laid down by the Banker, the number of Chances for the firft Card to be the Ponte's is « — I - M — 2 J alfo, that the number of Chances for the fecond to be the Ponte's, but not the firil, \% n — 2 x ;? 3 : wherefore the number of Chances for the firft to be the Ponte's, but not the fecond, is likewife « — 2 x n — 3. Hence it follows, that if from the 8o The Doctrine ©/"Chances. the number of Chances for the firft Card to be the Ponte's, vlx: from n — i x n — 2, there be fubtraded the number of Chances for the i^rft to be the Ponte's, and not the fecond, viz. n — 2 x « — 3, there will remain the number of Chances for both firft and fecond Cards ;o be the Ponte's, -viz. 2 x n — 2, and fo for the reft. Solution of the fourth Cafe. The number of Chances which the Banker has for winning and lofing, are as follow : I 2 3 4 5 6 7 8 \ n — 2 X n — 3 X n — 4 for winning i 3 X 71 — 2 X n — 3 for winning y n — 2 X n — 3 X n- — 4 for lofing r { ;/ — 4 X n — 5 X n — 6 for winning i 3 X ;/ — 4 X n — 5 for winning y n — 4 X n — 5 X n — 6 for lofing i { « — 6 X n — 7 X n — 8 for winning i 3 X n — 6 X n — 7 for winning y n — 6 X n — 7 X n — 8 for lofing i \ n — 8 X n — 9 X « — icfor winning i 3 X n — 8 X n — 9 for winning y n — 8 X n — 9 X n — i o for lofing i i 2 X 3 X 2 X 1 X 2 X I X ofor winning i 1 for winning y ofor lofing I Wherefore the Gain of the Banker, or the Lofs of the Ponte, is J_"~ ^_ fuppofing y to be := — . 2)7—5 or It will be eafy, from the general expreflions of the Loffes, to compare the difadvantage of the Ponte at Bafjette and Fharaon, under the fame circumftances of Cards remaining in the hands of the Banker, and of the number of times that the Ponte's Card is con- tained in the Stock; but to fave that trouble, I have thought fit here to annex a Table of the Gain of the Banker, or Lofs of the Ponte, for any particular circumftance of the Play, as it was done for BaJJette. A Ta- Ihe Doctrine o/' Chances. A Table for Pharaon. 8l N I 2 3 4 52 * * * * * ifr * * * *50 50 * * * 94 65 48 48 48 90 62 46 46 46 86 60 44 44 44 82 S7 42 42 40 42 40 78 74 54 52 40 38 38 38 70 49 36 36 36 66 46 34 34 34 62 44 32 32 32 58 41 30 30 30 54 38 28 28 28 52 36 26 26 26 45 33 24 24 24 42 30 22 22 22 38 28 20 20 20 34 2S 18 18 18 30 22 16 i6 16 a6 20 H 14 H 22 17 12 12 12 18 14 10 10 10 14 12 8 8 s II 9 6 1 The 82 TJoe Doctrine oj Chances. The numbers of the foregoihg Table, as well as thofe of the Tabic for Bafjettc, are fufficieiuly exad to give at firft view an idea of the advantage of the Banker hi all circumftances, and the Me- thod of ufing it is the fame as that which was given for Bafette. It is to be obferved at this Play, that the leaft diladvantage of the Ponte, under the fame circumftances of Cards remaining in the Stock, is when the Card of the Ponte is but twice in it, the next greater when three times, the next when once, and the greateft when four times.' O/' Perm UTAT IONS and Combinations. Permutations are tlie Changes whicli fevefal things can receive in the different orders in which they may be placed, being confidered as taken two and two, three and three, four and four, (sc. Combinations are the various Conjundtions which feveral things may receive without any refpe(5t to order, being taken two and two, three and three, four and four. The Solution of the Problems that relate to Permutations and Com- binations depending entirely upon what has been faid in the S'*" and q"" Articles of the Introdudion, if the Reader will be pleafed to con- fult thofe Articles with attention, he will eafily apprehend the reafon of the Steps that are taken in the Solution of thofe Problems. !P R O B L E M XV. Any number of things a, b, c, d, e, f, being given^ out of which two are taken as it happens : to fnd the Proba- bility that any of them^ as a, f^all be the firfi taken y and any other, as b, the fecond. Solution. The number of Things in this Exarhple being fix, it follows that the Probability of taking a in the firft place is ^\ \t\. a be confidered as taken, then the Probability of taking b will be - ; wherefore the Probability of taking <?, and th^n ^, is -^ x -- = I ■30* Coral- 75^ Doctrine <?/" Chances. 83 Corollary. Since the taking a in the firft place, and b in the fecond, is but one fingle Cafe of thofe by which fix Things may change their order, being taken two and two j it follows that the number of Changes or Permutations of fix Things, taken two and two, muft be 30. Univerfally -, let n l?c the number of Things ; then the Probability of taking a in the firft place, and b in the fecond will be -- x —- ; and the number of Permutations of thofe Things, taken two and two, will be « X « — J, PROBLEM XVI. Any number of Things a, b, c, d, e, f, being given ^ out of which three are taken as it happens ; to find the Probability that a pall be the firfl taken ^ b the fecond^ and c the third, Sol u t 10^. The Probability of taking a in the firft place is ^\\t\. a be confideied as taken, then the Probability of .taking b will be - : fuppofe now both a and b taken, then the Probability of taking c >yill be ^ : .wherefore the Probfibility of taking firft a, ^then ^, and thirdly c, will be -^ ;<- x - =;= -u— . CoR OLXA-RY. Since the taking a in the firft place, b in the fecond, and c in the third, is but one fingle Cafe of thofe by which fix Things may change their Order, being taken three and three ; it follows, that the num- ber of Changes or Permutations of fix Things taken three and three, muft be 6 x 5 x 4 = 120. Univerfally, if n hz the number of Things ; the Probability of taking a in the firft place, b in the fecond, and c in the third, will be — x X ; and the number of Permutations of n Things taken three and three, will be « x « — i x ;/ — 2. M 2 Gene- ^4 T'ke Doctrine (?/" Chances. General Corollary. The number of Permutations of n things, out of which as many jire taken together as there are Units in />, will be « x « — i x 11 2 X n — 3, &c. continued to fo many Terms as there arc Units in/. Thus the number of Permutations of fix Things taken four and four, will be 6x5x4x3:= 360, likewife the number of Permu- tations of fix Things taken all together will be6x5X4X3X2X 1 =720. PROBLEM XVII. To find the Probability that any number of things, whereof fome are repeated, fhall all be taken in any order pro- pofed : for inflame, that ^idihhhcQcc f jail be taken in the order wherein they are written. Solution. The probability of taking a in the firft place is ~ ; fuppofe one a to be taken, the Probability of taking the other is -| . Let now the two firft Letters be fuppofed taken, the Probability of taking b will be - : let this be alfo fuppofed taken, the Pro- bability of taking another b will be -| : let this be fuppofed taken, the Probability of taking the third b will be - ; after which there remaining nothing but the Letter c, the Probability of taking it be- comes a certainty, and confequently is exprefTed by Unity. Where- fore the Probability of taking all thofe Letters in the order given is 2 I 3 2 I r ' Corollary I. The number of Permutations which the Letters aabbbcccc may receive being taken all together will be ^ ' , ' ^ / \ \ \ = 1260. Corollary 2. The fame Letters remaining, the Probability of the Letters be- ing taken in any other given Order will be juft the fame as before : thus The Doctrine (^/'Chances. 85 thus the Probability of thofe Letters being taken in the prder cabaccchb will be — — - . 1260 General Corollary. The number of Permutations which any number n of Things may receive being taken all together, whereof the firft Sort is repeated p times, the fecond q times, the third r times, the fourth s times, &c. will be the Series «xn — ixw — 2X n — 3 y~ n — 4, &c. con- tinued to fo many Terms as there are Units \n p ^ qA^r oz n — s divided by the produd of the following Series, viz. p x p — i x p — 2, &c. q X q — I X q — 2, &c. r x r — i x r — 2, &c. whereof the firft muft be continued to fo many Terms as there are Units in ^, the fecond to fo many Terms as there are Units in q, the third to fo many as there are Units in r, &c. PROBLEM XVIIL Any number of Things a, b, c, d, e, f, being given : to find the Probability that in taking two of them as it may happen^ both a and hfhall be taken^ without any regard to order. Solution. The Probability of taking <z or ^ in the firft place will be ~ ; fuppofe one of them taken, as for inftance a, then the Probability of taking b will be — . Wherefore the Probability of taking both a and b will be 4 x — . ° s Corollary. The taking of both a and b is but one fingle Cafe of all thofe by which fix Things may be combined two and two ; wherefore the number of Combinations of fix Things taken two and two will be 6 <; — X — . I 2 Utiiverfally. The number of Combinations of n Things taken two and two will be — x -^^^ » PRO- 86 He Doctrine ^Chances, PROBLEM XIX. j^ny number of things a, b, c, d, e, f being given, to find the Probability that in taking three of them as they happen, they floall be any three propofed, as a, b, c, no refpedi being had to order. Solution. The Probability of taking either <?, or b, or r, in the firft place, will be 4- ; fuppofe one of them as a to be taken, then the Pro- bability of taking either ^ or c in the fecond place will be 4 • again, let either of them be taken, fuppofe b, .then the Probability of taking c in the third place will be ~ ; wherefore the Probability of taking the three things propofed, viz. «, b, r, will be ~ x t x - . .CU3 RO LiL A Ry. The taking of a, b, c, is but one finglc cafe of all thofc by which fix Things may be combined three and three ; wherefore the num- ber of Combinations of fix Things taken three and three will be 6 <; 4 „ _X^ X -=20. I A n Uni'ver/ally. The number of Combinations ofti things combined ac- cording to the number />, will be the fradion -~— ~ — " / " ^ " - ^ &c. both Numerator and Denominator 'being continued to fo many Term:^ as there are Units in p. PROBLEM XX. 'Tofnd what Probability there is, that in taking at ran- dom /even Counters out of twehe, 'whereof four are white and eight black, three of them fiall be white ones. Solution.. Ftrjl, Find the number of Chances for taking three white out of four, which will be-J-x-|x-r=:4' Secondly, 'Tlie Doctrine (?/'Chances. 87 Secondly, Find the number of Chances for taking four black out of eight : thefe Chances will be found tobe-x-x— x-^ = 70. Thirdly, Becaufe every one of the firft Chances may be joined with every one of the latter, it follows that the number of Chances for taking three white, and four black, will be 4 x 70 =r 280. Fourthly, Altho' the cafe of taking four white and three black, be not mentioned in the Problem, yet it is to be underftood to be implyed in it ; for according to the Law of Play, he who does more than he undertakes, is ftill reputed a winner, unlefs the contrary be exprefly ftipulated ; let therefore the cafe of taking four white out of four be calculated, and it will be found 7- x — x — x — 1234 Fifthly, Find the Chances for taking three black Counters out of o ^ £ eight, which will be found to be — x -^ x —r= c,6. Sixthly, Multiply the two laft numbers of Chances together, and the Produdl 56 will denote the number of Chances for taking four white and three black. And therefore the whole number of Chances, which anfwer to the conditions of the Problem, are 280 -\- 56 = 336. There remains now to find the whole number of Chances for taking feven Counters out of twelve, which will be -^ x -^ x 10 9 S 7 6 -— X-7X — x-^- X — = 702. Lajlly, Divide therefore 336 by 792, and the Quotient i^ 792 or — will exprefs the Probability required j and this Fradlion being fubtradled from Unity, the remainder will be -^ , and therefore the Odds againfl taking three white Counters are 19 to 14. Corollary. Let a be the number of white Counters, b the number of black, n the whole number of Counters ^= a -\- b , r the number of Coun- ters to be taken out of the number n ; let alfo p reprefent the number of white Counters to be found precifely in c, then the number of Chances for taking none of the white, or one fingle white, or two S 8 The Doctrine ^Chances. two white and no more, or three white and no more, or four white and no more, &c. will be expreffed as follows ; a a — I n — ? n — ^ b b—\ h — 2 . 1 X - X -^- X -—X — , 6cc. X y X -^ X -^, &c. The number of Terms in, which a enters being equal to tJhe number /», and the number of Terms in which h enters being equal to the number c — p. And the number of all the Chances for taking a certain number c of Counters out of the number «, is exprefled by the Series ^ X -^Y" • ■"""' ^ ~r' ' ^^* *° ^^ continued to as many Terms as there are Units in c, for a Denominator. Examples. Suppofe as in the laft problem ; only that of the 7 Counters drawn, there fhall not be one white. In this Cafe, fince /> = o, and c — p^=zjz=b — I : we are to take i of the firfl: Series, and 7 (or i ) Terms of the fecond ; which gives the number of Chances r x 8 i the Ratio of which to all the 7's that can be taken out of 12, is = — ; So that there is the Odds of o 8 to i, that there (hall 79- 99 ^ be one or more white Counters among the 7 that are drawn. Again, if there is to be i white Counter and no more, we are now t 1 rr-i " h b — ' l> — 2 b—7, b 4 to take the Terms i x — . . . x - x x x — - x — ^ I 1234s b~= 8 7 6 <; 4 % ■ ^ 8X7 112 X ~~ =4...x— X— X — X- X— x-=:4x r=. — : "^6 ^ 123456^2 Which gives the probability -^ = "TTi ^'^ ^^ '^^^^ ^5 ^^ \\\ that there (hall be more than i white Counter, or that all the 7 fhall be black. Lnjily, If it is undertaken to draw all the 4 white among the feven, the Number of Chances will be i x - x ^ x - = 56. And the Probability -^ = T" ' that is, the Odds of 92 to 7 that there fliall be, of the 7 drawn, fewer than 4 white Counters, or none at all. Remark. If the numbers n and c were large, fuch as n = 40000 and c = 8000, the foregoing Method would feem impradicable, by reafon of the vaft number of Terms to be taken in both Series, whereof the firft is to be divided by the fecond : tho' if thofe Terms were The Doctrine <?/'Chances. 89 were a(flually fet down, a great many of them being common Di- vifors might be expunged out of both Series; for which reafon it will be convenient to ufe the following Theorem, which is a contra<ftion of that Method, and which will be chiefly of ufe when the white Counters are but few. Let therefore n be the number of all the Counters ; a the number of white; c the number of Counters to be taken out of the number n ; p the number of the white that are to be taken precifely in the number c ; then making n — c^=ii. The Probability of taking pre- cifely the number fi of white Counters, will be c . c — I . c — '2, &c. X d . d — 1 . d—2, &c. X — X -^^^ x -^^^ &c 123' ***-• n . n — I . ?i — 2 . n — 3 . 7i — 4 . ?i — 5 . n — 6 . n — 7 . n — 8, &c. Here it Is to be obferved, that the Numerator confifts of three Series which are to be multiplied together; whereof the firft contains as ma- ny Terms as there are Units in p ; the fecond as many as there are Units in a—p ; the third as many as there are Units in p ; and the Deno- minator as many as there are Units in a. PROBLEM XXI. In a Lottery conftjiing o/' 40000 Tickets^ among which are three particular Benefits^ what is the Probability that taking 8000 of them^ one or fnore of the particular Be- nejits Jljall be amongfi them. Solution. F/r/?, In the Theorem belonging to the Remark of the foregoing Problem, having fubftituted refpedively 8000, 40000, 32000, 3 and I, In the room off, w, </, ^, and p ; it will appear that the Proba- bility of taking one precifely of the three particular Benefits, will be — ■■ ; = nearly. ' Secondly, c, n, d, a being interpreted as before, let us fuppofc p^r.2\ hence the Probability of taking precifely two of the par- ticular Benefits will be found to be '°"° ' "°°° " '''°^ " ^ =-11- 40000 . 39999 . 39998 ,25 nearly. N Thirdly, 90 7^^ Doctrine (t/* Chances. Thirdly, making /> = 3, the Probability of taking all the three particular Benefits will be found to be ■■ °^° ' '''^^ ' '^^ — ■. r=: - — r _ _ 40000 . 391)99 . i?99y!> 125' Wherefore the Probability of taking one or more of the three par- ticular Benefits will be - — , ''" or — — very near. It is to be obferved, that thofe three Operations might have been contraded into one, by inquiring the Probability of not taking any of the three particular Benefits, which will be found to be ,zopo. ,.9Q9.v9oH ^ j^ ^^^. ^j^.^j^ fubtraded from 40000 . 39999 . 199'-)'^ »*5 ^ b Unity, the remainder i ~ or —^ will {hew the Probability required, and therefore the Odds againft taking any of three particu- lar Benefits will be 64 to 6 1 nearly. PROBLEM XXII. To find how many Tickets ought to be taken in a Lottery confifiing of 40000, among which are Three particular Benefit Sy to make it as probable that one or more of thofe Three may be taken as not. SOLUTIOW. Let the number of Tickets requifite to be taken be = a: j It will follow therefore fi-om the Remark belonging to the xx*** Problem, that the Probability of not taking any of the particular Benefits will be "~"'- X "~2~i^ ' X " ~^~ ^ J but this Probability is equal to — , fance by Hypothefis the Probability of taking one or more of them is equal tp — , from whence we fhall have the Equation -— - x - -^ -■ ■ x — — — = - , which Equation being folved, the Value of x will be fouad to be nearly 8252. N. B. The Fadlors whereof both the Numerator and Denomi- nator are compofed, being but few, and in arithmetic progreflion ; and befides, the difference being very fmall in refpeft of n ; thofe Terms may be confidered as being in geometric Progreflion : where- fore the Cube of the middle Term "~^' , may be fuppofed MuaUo the produd of the Multiplication of thofe Terms; from whence iTje Doctrine o/" Chances. gt whence will arife the Equation -^=1^^ = - ; or, heglecfling U- li nity in both Numerator and Denominator, -^^^ — :=: — and confequently x will be found to be = « x i — V — or n x I — 7'*^4' but n =. 40000, and i — j V4 = 0.2063 ; wherefore a::=8252. In the Remark belonging to the fecond Problem, a Rule was given for finding the number of Tickets that were to be taken to make it as probable, that one or more of the Benefits would be ta- ken as not ; but in that Rule it was fuppofed, that the proportion of the Blanks to the Prizes was often repeated, as it ufually is in Lotteries : now in the cafe of the prefent Problem, the particular Benefits being but three in all, the remaining Tickets are to be confi- dered as Blanks in refpedl of them ; from whence it follows, that the proportion of the number of Blanks to one Prize being very near as 13332 to I, and that proportion being repeated but three times in the whole number of Tickets, the Rule there given would not have been fufficiently exadt, for which reafon it was thought neceffary to give another Rule in this place. PROBLEM XXIII. Suppojing a Lottery of 100000 Tickets^ whereof c^ooco are Blanks^ and 1 0000 are Benefits^ to determine ac- curately what the odds are of taking or not taking a Be- nefit^ in a?iy nutnher of 'Tickets affigned. Solution. Suppofe the number of Tickets to be 6 ; then let us inquire into the Probability of taking no Prize in 6 Tickets, which to find let us make ufe of the Theorem fet down in the Corollary of the xx* Problem, wherein it will appear that the number of Chances for ta- king no Prize in 6 Tickets, making a = loooo, b = 90000, c = 6, ^ = o, nr=z looooo, will be Qoooo Sqqqo SqqoS Roqo7 8qqQ6 8300; ; X X X — X — — X , and that the whole number of Chances will be loooro QgggQ qogqS qoQo- 99996 q999> l _-_ X — ^— X -.-r— X — ^ X —J- X —jr- i then N 2 dividing 92 Ihe Doctrine o/" Chances. dividing the firft number of Chances by the fecond, which may cafily be done by Logarithms, the Quotient will be 0.53143, and this ftitws the Probability of talcing no Prize in 6 Tickets : and this decimal fradion being fubtraded from Unity, the Remainder 0.46857 fliews the Probability of taking one Prize or more in 6 Tickets ; wherefore the Odds againft taking any Prize in 6 Tickets, will be 53143 to 46857. If we fuppofe now that the number of Tickets taken is 7, then carrying each number of Chances above-written one Hep farther, we (hall find that the Probability of taking no Prize in 7 Tickets is 0.47828, which fradion being fubtraded from Unity, the remainder will be 0.52 172, which fliews the Odds of taking one Prize or more in 7 Tickets to be 52172 to 47828. Remark. When the number of Tickets taken bear a very inconfiderabie proportion to the whole number of Tickets, as it happens in the cafe of this Problem, the Queftion may be refolved as a Problem de- pending on the Caft of a Die : we may therefore fuppofe a Die of 10 Faces having one of its Faces fuch as the Ace reprefenting a Benefit, and all the other nine reprefenting Blanks, and inquire into the Probability of miffing the Ace 6 times together, which by the Rules given in the Introdudion, will be found to be -j-^= 0.53144 differing from what we had found before but one Unit in the fifth place of Decimals. And if we inquire into the Probability of mifling the Ace 7 times, we fhall find it 0.47829 differing alfo but one Unit in the fifth of Decimals, from what had been found before, and there- fore in fuch cafes as this we may ufe both Methods indifferently ; but the firfl will be exad if we adually multiply the numbers together, the fecond is only an approximation. But both Methods confirm the truth of the pradical Rule given in our third Problem , about finding what number of Tickets are ne- cefTary for the equal Chance of a Prize ; for multiplying as it is there direded, the number 9 reprefenting the Blanks by 0.7, the Produd6.3 will fhewthat the number requifite is between 6 and 7. PROBLEM XXIV. T^he fame things being given as in the preceding Problem, fuppofe the price of each Ticket to be 10 -^- and that ajter the Lottery is drawn, 'j^- . — loP- be returned to 'The Doctrine <?/" Chances. 93 to the Blanks y to jind in this Lottery the value of the Cha?2ce of a Prize, Solution. There being 90C00 Blanks, to every one of which 7 ^- — 10 A is returned, the total Value of the Blanks is 675000 ^- and confe- quently the total Value of the Benefits is 325000 ^- which being divided by loooo, the number of the Benefits, the Quotient is 32 ^- — 10/'} and therefore one might for the Sum of 32 ^ — loJ^ be intitled to have a Benefit certain, taken at random out of the whole number of Benefits : the Purchafer of a Chance has there- fore I Chance in 10 for the Sum of 32^ — 10/^ and 9 Chances in I o for lofing his Money ; from whence it follows, that the value of his Chance is the lo''' part of 32 ^ \qJ" that is 3 ^ ^Jh- And therefore the Purchafer of a Chance, by giving the Seller 3 ^• • — 5/'- is intitled to the Chance of a Benefit, and ought not to return any thing to the Seller, altho' he fhould have a Prize j for the Seller having 3 ^ — 5^ fure, and 9 Chances in 10 for 7 ^ xofi- the Value of which Chances is 6 ^- — 1 5 >* j it follows that he has his I o ^• PROBLEM XXV. Suppojing fill the fame Lottery as has been mentioned in the two preceding Problems^ let A eftgage to furnifj B with a Chance^ on condition that whenever the Ticket on which the Chance depends, fall happen to be drawn, whether it proves a Blank or a Prixe, A fhall furnify B with a new Chance, and fo on., as often as there is occafon^ till the whole Lottery be drawn \ to find what confideration B ought to give A before the Lottery be- gins to be drawn, for the Chance or Chances of one or more Prizes, adtnitting that the Lottery will be 40 days a drawing. Solution. Let 3 ^ — 5/ , which is the abfolute Value of a Chance, be cal- led s. 94- T^l^^ Doctrine o/" Chances. r. A who is the Seller ought to cohfider, that the firfl: Day, he f urnilhes neccflarily a Chance whofe Value is i. 2°. That the fccond day, he does not neceflarily furnifli a Chance, but conditionally, iv'.s, if it lb happen that the Ticket on which the Chance depends, fliould be drawn on the firft day ; but the Proba- bility of its being drawn on the firft day is — ; and therefore he ought to take — 5 for the confideration of the fecond day. 3°, That in the fame manner, he does not neceflarily furnifh a Chance on the third day, but conditionally, in cafe the only Ticket depending (for there can be but one) fliould happen to be drawn on the fecond day ; of which the Probability being — , by reafon of the remaining 39 days from the fecond inclufive to the laft, it fol- lows, that the Value of that Chance is — s. Af". And for the fame reafon, the Value of the next is — i, and fo on. The Purchafer ought therefore to give the Seller I 4- -L 4- _L J- -1. J- -!- ^- i , the whole multiplied by j, or ' + 7 + T + 7 + T + 7 +-;^, the whok multiplied by j. Now it being pretty laborious to fum up thofe 40 Terms, I have here made ufe of a Rule which I have given in the Supplement to my Mifcellanea Analytka *, whereby one may in a very fhort time fum up as many of thofe Terms as one pleafes, tho' they were j 0000 or more j and by that Rule, the Sum of thofe 40 Terms will be found to be 4.2785 very near, which being multiplied byi which in this cafe is 3.25, the produdl 13.9 will fhew that the Pur- chafer ought to give the Seller about 13^ — 1 8 v^- COROLLR AV. The Value of the Chance s for one fingle day that fliall be fixed upon, is the Value of that Chance divided by the number of Days intercepted between that Day inclufive, and the number of Days re- maining to the end of the Lottery : which however muft be under- flood with this reftridion, that the Day fixed upon mufl be chofe be- fore the Lottery begins j or if it be done on any other Day, the State of 7:5d Doctrine o/" Chances. 95 of the Lottery muft be known, and a new Calculation made accord- ingly for the Value of s, * SCHOLIUfiJ. If there is a Series of Fradions of this Form ^ + -j^jrp 4" -J \ 1 1 !— A 1— . ; the firft of which is n-lfZ '«-l-3'a+4 'a — i' ^ , an4 the laft ' a—x ' > ^^^^ ^^^ w^^^ ^s» log, -4--i-J — L A + 4rB4--^, C-f -^D + ficc; ^JL 4._LA4--LBH--VC-h^D4-&c. to be obferved, 1". That the mark {log) denoting Nepers^ or the Natural Loga- rithm, affeds only the firft Term — . 2*. That the Values of the Capital Letters are, A = - , B = numbers of Mr. James Bernoulli in his excellent Theorem for the Summing of Powers ; which are formed ffQpa t$^\\, pther U fr>UpW6 s B— 1 — -_-i A C:=4~4--A-'— !-*B. 3-4 3.4 z 73'. 4 .5 . 6^" 3-4 2. 3-4-S-6 a. 3. 4. 576:7:^*^ 272 2 D^-L-l--iA— ^ 292 z p I ^^ I "'a "^ • "~ j ' 17 "^"2 2 &c. 3°. Ih woricing by this Rule, it will be convenient to fum a few of the firft terms, in the common way ; that the powers of — may the fooner converge. 4°. The l^me Rule furnifl^s an eafy Computation of the Loga- ■rithm of any ratio - , the difference of whofe terms is not very gre^t. PRO' 96 He Doctrine 0/" Chances. PROBLEM XXVI. 'To find the Frohability of takiiig four Hearts^ three Dia- monds , two Spades J and one Club inteji Cards out of a Stock CQJitainhig thirty-two. Solution. Firjl, The number of Chances for taking four Hearts out of the whole nuin bcr of Hearts that are in the Stock, that is out of Eight, Will be ■ ' - ^ • "^ • ^ = 70. Secondly t The number of Chances for taking three Diamonds out of Eight, will be ^^-^ = 56. Thirdly, The number of Chances for taking two Spades out of Eight, will be \ ' I =28. Fourthly, The number of Chances for taking one Club out of Q Eight, will be — = 8. . And therefore multiplying all thofe particular Chances together, ^^ the produdl 70 x 56 x 28 x 8 t^ 878080 will denote the whole num- ber of Chances for taking four Hearts, three Diamonds, two Spades, and one Club. Fifthly, The whole number of Chances for taking any ten Cards out of thirty-two is . ■ 2 . 3 . 4 ■ s ■ 6 . 7 • « • 9 . 10 =04512 240. And therefore dividing the firfl Produ6t by the fecond, the quotient "^ 1°^^ P or — ^ nearly, will exprefs the Probability required ; from which it follows that the Odds againft taking four Hearts, three Diamonds, two Spades, and one Club in ten Cards, out of a Stock containing thirty-two, are very near 74 to i . Remark. But if the numbers in this Problem had not been reftrift- ed each to a particular fuit of Cards j that is, if it had been under- taken only that in drawing the ten Cards, 4 of them fhould be of one fuit, 3 of another, 2 of another, and one of the fourth ; then writing for the four fuits, the Letters A . B . C . D ; and under them the Numbers 4 . 3 . 2 . 13 fmce this is 7he Doctrine o/" Chances. 97 is but one Pofition out of 24, which the numbers can have with refpedl to the Letters (by the general Corollary to Prob. xvi) we muft now multiply the number of Chances before found, which was 878080, by 24; and the probability required will be "^7-777^ J that is, it is the Odds of about 2 to i, or very nearly of 68 to 33, that of 1 o Cards drawn out of a Piquet pack four^ three, two, and one, fhall not be of different fuits. Of the Game of Q^u a d r i l l e. PROBLEM XXVII. The Player having 3 Matadors and three other Trumps by the lowefi Cards in black or red^ what is the Pro- bability of his forcing all the Trumps f Solution. In order to folve this Problem, it is to be confidered, that the Player whom I call A forces the Trumps neceflarily, if none of the other Players whom I call B, C, D, has more than three Trumps ; and therefore, if we calculate the Probability of any one of them having more than three Trumps, which cafe is wholly againfl A, we may from thence deduce what will be favourable to him ; but let us firft fuppofe that he plays in black. Since the number of Trumps in black is 1 1, and that A by fup- pofition has 6 of them, then the number of Trumps remaining amongft 5, C, Z) is 5 ; and again, fince the number of all the other remaining Cards, which we may call Blanks, is 29, whereof A has 4, it follows that there are 25 Blanks amongft B, C, D ; and there- fore the number of Chances for B in his i o Cards to have 4 Trumps and 6 Blanks, is by the Corollary of the xx'"" Problem. And likewife the number of Chances for his having 5 Trumps and 5 Blanks, is by the fame Corollary. X 24 . 2? O And ^8 The Doctrine oj Chances. A-h^ therefore the number of all the Chartces of B againft A is 106 X 5 X 7 X 1 1 X 23 : but the number of Chances whereby any 10 Cards may be taken out of -zo is ^°- ^9 • ^8 . 2- . ;6 . z^ . 24 . 2? . 2z . 2. Which being reduced to 5 x 7 x 9 x n x 13 ^ 23 x 29, it follows that the Probability of B's having more than three Trumps is 106 X S .7 ■ J I ■ 23. 106 1 . .r. , f-.. 3.3.5.7.11 . 13 . 2^ 29 — 9 >; ,3 . 2g : but this Probability falls as well upon C and D as upon B, and therefore it ought to be multiplied by 3, which will make it ^ ^ ,'°^^ ^^ = ~ . and this being fubtradled from Unity, the remainder -^^^ ^[\\ exprefs the Probabihty of jfs forcing all the Trumps ; and therefore the Odds of his forcing the Trumps are 1025 againft 106, that is 29 to 3 nearly. But if y^ plays the fame Game in red, his advantage will be con- fiderably lels than before ; for there being 1 2 Tfumps in red, whereof he has 6, B may have 4, or 5, or 6 of them, fo that the number of the Chances which B has for more than three Trumps will be Tefpe<SiveIy as follows : 6^ ; X 4- . 3 24 ■> 23 . 22 . 21 . 20 . 19. "1-2.3.4 1 . 2 . 3 . '4 . s . 6 6 . 5 . 4. . 3 . 2 24. . 2g 22..: 2L . 20 J. 2. 3. 4. 5 1.2.3.4 5... 4.3.2.1 -24 . 23 . 22 ■ 21 1.2.3.4.5.0 I .2.3.4 Now the Sum of all thofe Chances being 215 x 23 x 22 x 21, and the Sum of all the Chances for taking any 10 Cards out of 30, being 5 x 7 x 9 x ii x 13 x 23 x 29, as appears by the preceding cafe, it follows, that the Probabihty of B's having more than three T r^mps IS- . . ,.g. ,. . ,3 . .3 . 29 - — 3 ■ ■3.^9 ' ' ^"^ ^^'' Pro- bability falling as well upon C and D, as upon B, ought to be mul- tiplied by 3, which will make it ,^ _\^ = — ; and this being fubtra<fted from Unity, the remainder -— — , will exprefs the Pro- bability of A's forcing all the Trumps ; and therefore the Odds of liis forcing all the Trumps is in this cafe 291 to 86, that is nearly 10 to 3. PRO- iToe Doctrineo/"Chances. 99 PROBLEM XXVIII. 7^^ Player A having Spadille^ Manille, Kingy ^een, and two /mail Trumps in blacky to jijid the Probability of his forcing all the Trtwips. Solution. A forces the Trumps neceflarily, if Bafte accompanied with two other Trumps be not in one of the Hands of B, C, D, and as BaJIe ouglit to be in fome Hand, it is indifferent where we place it ; let it therefore be fuppofed that B has it, in confequence of which let us conllder the number of Chances for Ijis having be- fides Bajie, 1°. 2 Trumps and 7 Blanks. 2°. 3 Trumps and 6 Blanks. 3°. 4 Trumps and 5 Blanks. Now the Blanks being in all 29, whereof A has 4, it follows that the number of remaining Blanks is 25 ; and the number of Trumps being in all 11, whereof A has 6 by Hypothefis, and B has I, viz BafiCy it follows that the number of remaining Trumps is 4 ; and therefore the Chances which B has againft the Player are refpedively as follows : 1.2. 3.2.1 3 X X X z--, . 24 • 23 22 . 21 . 20 . «9 1 . 2 ^\ . 24 3 • 23 4 • . 22 . 5 • 21 6 . . 20 7 I . 2 25 . 24 1 . 2 • 3 • 3 4 • . 22 . 4 5 21 5" 6 The Sum of all which is 1441 x 5 x 23 x 22 ; but the Sum ol all the Chances whereby B may join any 9 Cards to the Bafte which 1 1 1 1 • 2q X 2S X 27 . 26 . 2C . 24 . 2T X C2 . 2 1 ,he has already is f . , . 3 . 4—/— 6 - 7 T 3-9" — 29 x 7 x 3 x 13x5x23x11. and therefore the Probability of Bafie being in one Hand, accompanied with t,wo Trijmps at leaft, is exprefled L ..L -C n.- M4' • 5 ■ 23 • 2 ^ ' ' n: . 22 '882 by thePraftion ,, . , . . . .3. ;, ,,,-,, — .^.;., ., = _t^ and this being fubtraded from Unity, the remainder will be -^Prr » and therefore the Odds of yf' s forcing the Trumps are 5035 to 2882, which are very near 7 to 4. O 2 But I CO I'he Doctrine o/* Chances. But if it be in red, A has the fmall difadvantage of 19703 againft 19882, or nearly no againfl iii. It is to be noted in this Propofition, that it is not now neceflary to multiply by 3 ; by reafon that B reprefents indeterminately any one of the three B, C, D : elfe if the cafe of having Ba/ie was deter- mined to B in particular, his probability of having it would only be — : fo that the Chances afterwards being multiplied by 3, the So- lution would be the fame. PROBLEM XXIX. The Player having Spadille^ Manille^ and 5 other Trumps more by the loweji in red^ what is the Probability^ by playing Spadille and Manille, of his forcing 4 Trumps f Solution. The 5 remaining Trumps being between B^ C, £), their various difpofitions are the following : 5, C, D I, 2, 2 2, 3. 3. I, I 4, I. 5. 0, Which muft be underftood in fuch manner, that what is here afligned to B may as well belong to C or JD. Now it is plain, that out of thofe five difpofitions there are only the two firft that are favourable to A-, let us therefore fee what is the Probability of the firft difpofition. The number of Chances of B to have 1 Trump and 9 Blanks C 2C 24 . 2^ . 22 . 21 :0 iq 18 !■' are 7 X . . 2 . 3 . 4 ■ s ■ '■ — 7 ■ « ■ 9 ^S^ S^ S^''^ ^^7 X 19 X 23, but the number of all the Chances whereby he may take any 10 Cards out of 30, is 5 x 7 x 9 x 1 1 x 13 x 23 x 29 as has been feen already in one of the preceding Problems ; and therefore the Probability of J5's having one Trump and nine Blanks is <: • ? • 5 • " • ' 7 • '9 • '3 25 ■ '9 • '7 5 . 7 . 9 . >i . 13 . 23 . 29 7 • 9 • »3 • 29 * Now 'The Doctrine o/"Chances. ioi Now In order to find the number of Chances for C to have 2 Trumps and 8 Blanks, it muft be confidered that A having 7 Trumps, and B i, the number of remaining Trumps is 4 ; and likevvife that A having 3 Blanks, and B 9, the number of remaining Blanks is 16, and therefore that the number of Chances for C to have 2 Trumps^ and 8 Blanks is 4 . ? 16 . I C . 14. . I ? . 12 . I I . 10 . Q /• T-.-7 X . . . -. 3 — 4 T 5— 6-^— 8 = 6 X 9 X 10 X 1 1 X 13^ But the number of all the Chances whereby C may take any 10 Cards out of 20 remaining between him and D, is 20. 10 . 18 . 17 . 16 . i; . 14. 13 . 12 . ti — :; i — 7 = 4x11x13x17x10, I. z. 3. 4. 5. 6. 7. 8. 9. 10 ^ •> I y* and therefore the Probability of Cs having 2 Trumps and 8 Blanks is 6 . Q , 10.11 . 13 6 . 9 . 10 9 ■ I ; 4 . II . 13 . 17 . 19 4 ■ 17 • >9 »7 • '9 Now A being fuppofed to have had 7 Trumps, B r, and C 2, D muft have 2 necefl'arily, and therefore no new Calculation ought to be made on account of D. It follows therefore that the Proba- bility of the difpofition i, 2, 2, belonging refpedively to J?, C, Z), ought to be expreffed by — ^ ' "^ ' ' ' — x '^ = — ^— — ^— — . & r -^ 7 . 9 . 13 . 29 17 • '9 7 ■ 13 • 9 Now three things, whereof two are alike, being to be permuted 3 different ways, it follows that the Probability of the Difpofition I, 2, 2, as it may happen in any order, will be ^ '^ ^\^ \ ^l =^ ItT^' It will be found in the fame manner, that the Probability of the Difpofition 2, 3, o as it belongs refpedlively to B, C, D, is - ' • '' ' '° : but the number of Permutations of three things which 7.13.29' " are all unlike being 6, it follows that the faid Probability ought to be multiplied by 6, which will make it — -^ — / ■ — — = ^^ ' . From all which it follows, that the Probability of A's forcing 4 Trumps is "^^^^' '"° - = 4f^ * which fradion being fubtradted from Unity, the remainder will be -2ii- , and therefore the Odds . 2-39 of -<fs forcing 4 Trumps are 1725 to 914, that is very near 17 to 9. p R a> I02 Ithe Doctrine o/" Chances. PROBLEM XXX. A the "Player having 4 Matadors^ in Diamonds^ with the two black Ki?igs^ each accompanied with two fmcll Cards of their own fuit ; what is the Probability that no one of the others B, C, D, has more than 4 Trumps, or in cafe he has more, that he has alfo of the fuit of both his Kings ', in which cafes A wins neceffarily f Solution. The Chances that are againft A are as follows ; it being poffi- ble that B may have Diamonds, Hearts, Number of Chances. 5, 5 14112 «, 4 5880 7» 3 960 "j 2 Sum 45 20997 Diamonds Spades, Hearts, Number of Chances. 5» I 4 70560 S» 2 3 100800 5» 3 2 50400 5> 4 I 8400 5» 5 , 336 6. I 3 , 20160 6, 2 2 18900 6, 3 1 5600 6, 4 1 420 7. I 2 2 160 7. 2 I 1200 3 160 o» I I 60 8, 2 15 I 279171 Now 7^^ Doctrine o/" Chances. 103 Now by reafon that among B, C, D, there are as many Clubs as Spades, viz.. 6 of each fort, it follows that the Clubs may be fub- flituted in the room of the Spades, which will double this kft num- ber of Chances, and make it 558342 ; and therefore, adding coge- ther the firft and fecond number of Chances, the Sum will be 579339. which will be the whole number of Chances, whereby S may withfland the Expeaation of A; but the number of all the ■Chances which B has for taking any 10 Cards out of 30, is 5 x 7 x 9x11x13x23x29 = 300450 1 5 ; from which it follows that the Probability of B's withftanding the Expedation of ^ is 30-4^0.5 * but as this may fall as well upon C and D as upon B, it follows that this Probability ought to be maltipiied by 3, then the Produdt Ico^^^^^ will exprefs the Probability of A's lofingi and this being fubtradled from Unity, the remainder will exprefs the Probability of ul's winning ; and therefore the Odds of As winning will be little more than 16 — to i. 4 PROBLEM XXXI. A havtfjg Spadilk^ Manilk^ King, K?m've, and tW9 other fmall Trumps in black, what is the Probability that Bajle accompanied with two other Trumps, or the ilueen accompanied with three other, p.mll not be in the fame hand \ in which cafe A wins necej[arily P Solution. The Probability of Bafle being in one hand, accompanied with two other Trumps, has been found, in Problem xxviii, to be " ■ . The number of Chances for him who has the Queen, to have alfo three other Trumps, excluding Bafte, is 2 A 21 22 '*! '*'^ ^ ^ ■ • " r=r 5 X7XIIX20X23 but the number of Chances for joining any 9 Cards to tlie Queen is 3 X 5 X 7 X 1 1 X 13 X 23 X 29, and therefore the Probability of the Queen's being in one hand, accompanied with three other Trumps, is y . 7 . 1 1 ■ 20 3^ ;o 20 1 10 jx 5 . 7 . II . 13 . 23 . ^9 3.13. 29 1131 7917 ' now 104 ^^ Doctrine <j/'Chances. now this Probability being added to the former, the Sum will be ■ ^°" ; and therefore the Odds of yf s not being withftood either 7917 ' o from Bafte being accompanied with two other Trumps, or from the Queen accompanied with three, are 4895 to 3022, nearly as 13 to 8. It may be obferved, that the reafon of Bafte being excluded from among the Trumps that accompany the Queen is this ; if the Queen be accompanied with Bafte and two other Trumps, the Bafte itfelf is accompanied with three Trumps, which cafe had been taken in already in the firft part of the Solution. PROBLEM XXXII. A havinv three Matadors in Spades with the Kings of Heart Sy Diamonds^ and Clubs, two fmali Hearts, and two [mall Diamonds \ to find the Probability that not above three Spades pall be in one hand, or that, if there be above three, there pall be alfo of the fuits of the three Kings ; in which cafe A wins 7tecejfarily, Solution. The Probability of not above ^three Trumps being in one hand = 0.332141. The Probability that one of the Oppofers fliall have 4 Trumps, and at the fame time Hearts, Diamonds, and Clubs, and that no other fliall have 4 Trumps, is =0.393501. The Probability that two of the Oppofers fliall have 4 Trumps, and at the fame time Hearts, Diamonds, and Clubs, is =: 0.013836. The Probability that one of the Oppofers fliall have 5 Trumps, and at the fame time Hearts, Diamonds, and Clubs, is = 0.103019. The Probability that one of the Oppofers fliall have 6 Trumps, Hearts, Diamonds, and Clubs, 15=^0.001041. The Probability of one of the Oppofers having 7 Trumps, and at the fame tiinc Hearts, Diamonds and Clubs, is = 0.0003 13. Now the Sum of all thefe Probabilities is 0.843851, which being fubtradted from Unity, the remainder is 0.156149 ; and therefore the Odds of the Player's winning are as 843851 to 156 149, that is very near as 27 to 5. PRO- Th Doctrine ^Chances. 105 PROBLEM XXXIII. 'To find at Pharaon, how much it is that the Banker gets per Cent, of all the Money that is adve?itured. Hypothesis. I fuppofe frjl^ that there is but one fingle Ponte j Secondly^ that he lays his Money upon one fingle Card at a time ; Thirdly, that he begins to take a Card in the beginning of the Game ; Fourthly, that he continues to take a new Card after the laying down of every couple : Fifthly y that when there remain but fix Cards in the Stock, he ceafes to take a Card. Sol u T ION. When at any time the Ponte lays a new Stake upon a Card taken as it happens out of his Book, let the number of Cards already laid down by the Banker be fuppofed equal to x, and the whole number of Cards equal to «. Now in this circumftance the Card taken by the Ponte has pafl four times, or three times, or twice, or once, or not at all. F/r/?, If it has pafTed four times, he can be no lofer upon that account. Secondly, If it has pafTed three times, then his Card is once in the Stock : now the number of Cards remaining in the Stock being n — X, it follows by the firfl cafe of the xiii"' Problem, that the Lofs of the Ponte will be -;—- : but by the Remark belonging to the xx**" Problem, the Probability of his Card's having pafTed three times precifely in x Cards is ^ ,^_ ,' ' ^Z! IZ. • "o^"' ^"P- pofing the Denominator equal to S, multiply the Lofs he will fuitcr, if he has that Chance, by the Probability of having it, and the produa - • '~' ^/~' "^ will be his abfolute Lofs in that cir- cumflance. Thirdly, If it has pafTed twice, his abfolute Lofs will, by the fame Way of reafoning, be found to be • "' • ' ~ ' - -•' — ^+" ^ Fourthly, If it has pafl'ed once, his abfolute Lofs will be found to be Fijtbly, r — X . n — X — - C . ?K — ?.r "^ ^ 2S' Now the Sum of all tl: 2 ;+5« — 3* — 2 + 3^5 io6' The. Doctrine <?/* Chances. Fifthly, If it has not yet paflec^, his abfolute Lofs will be Loffes of the Pontc will be ^ , and this is the Lofs he fuffers by venturing a new Stake after any number of Cards x are pafTed. But the number of Couples which at any time are laid down, is always one half of the number of Cards that are paffed ; where- fore calling t the number of thofe Couples, the Lois, of the Ponte a! — — nn->r^n — 6t — 6tt-{-z^t may be exprefled thus -^ Let now p be the number of Stakes which the Ponte adven- tures ; let alfo the Lofs of the Ponte be divided into two parts, viz. and And fince he adventures a Stake p times ; it follows that the firft part of his lofs will be j- In order to find the fecond part, let / be interpreted fucceffively by o, I, 2, 3, &c. to the laft term p — i ; then in the room of bt we fhall have a Sum of Numbers in Arithmetic Progreffion to be multiplied by 6 ; in the room of btt we fliall have a Sum of Squares, whofe Roots are in Arithmetic Progreffion, to be multi- plied by 6 ; and in the room of 24^^ vve fhall have a Sum of Cubes,, whofe Roots are in Arithmetic Progreffion, to be multiplied by 24. Thefe feveral Sums being coUedled according to the Rule given in the fecond Remark on the x''' Problem, will be found to be (,._,,p^ + fft. + 2p ^^^ therefore the whole Lofs of the Ponte will be 6 pni — —p„„ + 5/»»4- 6/+ — 1 4/» H- 6pp + zp 2 z " 5 Now this being the Lofs which the Ponte fuftains by adven- turing the Sum p, each Stake being fuppofed equal to Unity, it follows that the Lofs per Cent, of the Ponte, is the quantity above- written multiplied by 100, and divided by />, which confidering that S has been fuppofed equal to « x n — i x n — 2x n — 3, will 1 2.n — c \ p — J y 6pp — 8/> — 2 .^ ,_^ make It to be 'f-^'jrTTTir^ ^ ^^"^ + • ,.«-;, .-a .,.-3 ^ ^°° ' let The Doctrine <?/"Chances. 107 let now n be interpreted by 52, and /> by 23 ; and the Lofs^^r Cent. of the Ponte, or Gain per Cent, of the Banlcer, will be found to be 2.99251 ; that is 2^ — 19/^ — lO'^ per Cent. By the fame Method of procefs, it will be found that the Gain per Cent, of the Banker at Bajfette will be - — „^_"~'^„_^ x 100 4- '^^ ^~ '_ ^~^_ ^ io°- Let K be interpreted by 5 1 , and /> by 23 ; and the foregoing expreffion will become 0.790582 or 15^^ — 9 ~" • The confideration of the firft Stake which is adventured before the Pack is turned being here omitted, as being out of the gene- ral Rule ; but if that cafe be taken in, the Gain of the Banker will be diminifhed, and be only 0.76245, that is \^fi- — 3 '^- very near; and this is to be eftimated as the Gain per Cent, of the Banker, when he takes but half Face. Now whether the Ponte takes one Card at a time, or feveral Cards, the Gain per Cent, of the Banker continues the fame : whe- ther the Ponte keeps conftantly to the fame Stake, or fome time doubles or triples it, the Gain per Cent, is flill the fame : whether there be one fingle Ponte or feveral, his Gain per Cent, is not thereby altered. Wherefore the Gain per Cent, of the Banker, upon all the Money that is adventured at Pbaraon, is 2^' — 19/" — lo'' and at Bajfette 15 / 3 '^• PROBLEM XXXIV. Suppojifig A and B to play together -, that the Cha?ices they have refpeEiively to wm are as a to b, and that B obliges himfelf to fet to Kfo long as A wins without in- terruption : what is the advantage that A gets by his hand f Solution. Firfl^ If A and B ftake i each, the Gain of A on the firft Game is Secondly, His Gain on the fecond Game will aUo be ^T^' , provided he (hould happen to win the firft : but the Probability of A's winning the firft Game k -4— . Wherefore his Gain on the fecond Game will be —^-— x "~^ . . a-f- b a-\-b P 2 Thirdly, io8 Tl^e Doctrine ^Chances. Thirdly^ His Gain on the third Game, after winning the two firft, will be likewife -J^^ : but the Probability of his winning the two firft Games is ====rr j wherefore his Gain on the third " + /)•• Gna a — b ame is ■->. x . u ■'f by a + d Fourthly, Wherefore the Gain of the Hand of A is an infinite Series : viz. i -\- — ;-— J- — ■ ■. i _, 4_ ■ &-c to be multiplied by ^~^ : but the Sum of that infinite Series is -^4^ J wherefore the Gain of the Hand of ^ is -^-^ x ""^ = a — 6 b Corollary i. HA has the advantage of the Odds, and B fets his hand out, the Gain of -^is the difference of the numbers expreffing the Odds, divided by the leffer. Thus if A has the Odds of 5 to 3, then his Gain will be ''~- - = — . 3 3 Corollary 2. If B has the difadvantage of the Odds, and A fets his hand out, the Lofs of B will be the difference of the numbers expreffing the Odds divided by the greater : thus if B has but 3 to 5, his Lofs will bei-. 5 Corollary 3. If A and B do mutually engage to fet to one another, as long as either of them wins without interruption, the Gain of A will be found to be ■— ^^ — ; that is the Sum of the numbers expref- fing the Odds multiplied by their difference, the Produdt of that Multiplication being divided by the Produdt of the numbers expref- fing the Odds. Thus if the Odds were as 5 to 3, the Sum of 5 and 3 being 8, and the difference being 2 , multiply 8 by 2, and divide the produft 16, by the produdt of the numbers expreffing the Odds, which is 1 5, and the Quotient will be -y- , or i -— - , which therefore will be the Gain of A. Corollary 4. But if he be only to be fet to, who wins the firft time, and that he be to be fet to as long as he wins without interruption ; then the Gain 7%e Doctrine ^Chances. 109 Gain oi A will be "'"— : thus if a be 5, and ^ 3, the Gain of ^ will be _^ = -f2-. 120 DO PROBLEM XXXV. Any number of Letters a, b, c, d, e, f, &*€. all of them different, bei?ig taken promifciwufly as it happens : to find the Probability that fome of thejn Jhall be fou72d in their places according to the rank they obtain in the Alpha- bet \ and that others of them JJjall at the fame time be difplaced. Solution. Let the number of all the Letters be = « ; let the number of thofe that are to be in their places be =/>, and the number of thofe that are to be out of their places = q. Suppofe for brevity's fake i_ I I I n ' n . n — I ' n . n — i . n — 2 ' n . n — I . n — 2 . n — 3 z=zv, &c. then let all the quantities i,r, s, /, v, &c. be written down with Signs alternately pofitive and negative, iDeginning at i, if /» be r= o ; at r, if /> be = I : at s, if p be = 2, &c. Prefix to thefe Quantities the Coefficients of a Binomial Power, whofe index is = q ; this being done, thofe Quantities taken all together will ex- press the Probability required. Thus the Probability that in 6 Letters taken promifcuoufly, two of them, viz. a and b fliall be in their places, and three of them, viz. c, d, e, out of their places, will be __; 3 J I ; " 6.5 6.5.4~r 6.5.4.3 0.5.4.3.2 720 And the Probability that a fhall be in its place, and b, c, d, e, out of their places, will be _i. __ ^ \ 6 4 J I "^S 6 t>.5~i~h.5.4 6.5.4.3 16. 5. 4. 3. 2 720 The Probability that a fhall be in its place, and b, c, d, e,f, out of their places, will be "T"o.5.4 O.5.4.3I0.5.4. 6.5 ' 0.5.4 <^S'4-3 ' 0.5.4.3.2 0.5.4.3.2.1 44 " 720 ibe The no Tl^e Doctrine o/" Chances. The Probability that c, ^, f , </, f , / {hall all be difplaced is 6^ i_ 1 ? 20 , I c 6 ^ 6 "''6.5 (> . 5 -4 "■" t'-5-4-3 6.S-4-3-2 + ' I < ■ I < I I ' 0.5.4.3.2.1 I2 bl24 120' 7Z0 g6c; __ 53 720 ■ 144 Hence it may be concluded, that the Probability that one of them at leaft fhall be in its place, is i ^ -|- 4- — -^ — 4" — ! ^— = -^^ , and that the Odds that one of them at 120 720 144 leaft fliall be fo found, areas 91 to ^'>^. It muft be obferved, that the foregoing Expreflion may ferve for any number of Letters, by continuing it to fo many Terms as there are Letters: thus if the number of Letters had been feven, the Probability required would have been -^ . Demonstration. Tlie number of Chances for the Letter a to be in tlie firft place, contains the number of Chances by which a being in the firft place, 6 may be in the fecond, or out of it : This is an Axiom of common Senfe of the fame degree of evidence, as that the whole is equal to all its parts. From this it follows, that if from the number of Chances that there are for a to be in the firft place, there be fubtradted the num- ber of Chances that there are for ^ to be in the firft place, and 6 at the fame time in the fecond, there will remain the number of Chances by which a being in the firft place, l> may be excluded the fecond. For the fame reafon it follows, that if from the number of Chances for a and l> to be refpeftively in the firft and fecond places, there be fubtrad:ed the number of Chances by which a, i>, and c may be refpedtively in the firft, fecond, and third places ; there will remain the number of Chances by which a being in the firft, and b in the fecond, c may be excluded the third place : and fo of the reft, hi -\- a denote the Probability that a fliall be in the firft place, and let — a denote the Probability of its being out of it. Likewife let the Probabilities that i> (hall be in the fecond place, or out of it, be refpecSively expreft by -}- (^ and — if. Let The Doctrine <7/Chances. hi Let the Probability that a being in the firft place, b fhall be in the fecond, be exprefTed by a ■\- b : Likewile let the Probability that a being in the firft place, b fhall be excluded the fecond, be exprefTed by a — b. Univerfally. Let the Probability there is that as many as are to be in their proper places, (liall be {o, and that as many others as are at the fame time to be out of their proper places fhall be fo found, be denoted by the particular Probabilities of their being in their proper places, or out of them, written all together : So that, for in- ftance a ■\- b •\- c — d — e, may denote the Probability that <?, ^, and c (hall be in their proper places, and that at the fame time both d and e fhall be excluded their proper places. Now to be able to derive proper Conclufions by virtue of this No- tation, it is to be obferved, that of the Quantities which are here confidered, thofe from which the Subtradlion is to be made are in- differently compofed of any number of Terms conneded by -|- and — ; and that the Quantities which are to be fubtradled do exceed by one Term thofe from which the Subtradlion is to be made -, the reft of the Terms being alike, and their Signs alike ; then nothing more is requifite to have the remainder, than to preferve the Quan- tities that are alike, with their proper Signs, and to change the Sign of the Quantity exceeding. It having been demonflrated in what we have faid of Permutations and Combinations, that a^=. — •. a-\-b:= -» — ' ; a A- b ■\- c^=. ^ , &c. let — , ! , &c. be refpedlively called r, s, /, V, &c. this being fuppofed, we may come to the following Conclufions. b =r b -^ a ^= s then 1°. b — a^=r — s c ■\- b Trr i for the fame reafon that a -^ b^=: s. _cj\- b -{- a^t 2*. 7~4- ^ — a^=s — ; f — a =r — s by the firft Conclufion. c —. a ■]- b = -\- s — ^by the fecond. c — a — b ^=f — 2s-\-t 3 'd-{' c -}- b = t d ■}- c -\- b -\-a:=v A.'>. d -i- c -i^b — a = t — v ■ ^-h^ 112 7^^ Doctrine of Chauces, d -Yc — a =^s — t by the fecond Conclufion. d -\-c — a -r- i' = / — -u by the fourth. 5°. ^ 4"^ — ^ — ^ = J 2/-|- V ~d —b ^ a z= r— 25-]- / by the third Conclufion. d —b — a -\- c ■= s —zt -f- *" by the fifth. By the fame procefs, if no letter be particularly afTigned to be in its place the Probability that fuch of them as are afligned may be out of their places, will likewife be found thus. — ar= I — r, for 4" ^ and — a together make Unity. — a -]^ b ^=r — shy the firft Conclufion. 7". — a — ■ b ■= I — 2r •\- s — a — b r=z I — 2r -\- s by the feventh Conclufion. — a — b •\- c = r — 2s -}- t hy the third. 8". — a — b — c = I — S'" -j- 3-5 — A Now examining carefully all the foregoing Conclufions, it will be perceived, that when a Queftion runs barely upon the difplacing any given number of Letters, without requiring that any other fhould be in its place, but leaving it wholly indifferent ; then the Vulgar Algebraic Quantities which lie at the right-hand of the Equation?, be<^in conftantly with Unity ; it will alfo be perceived, that when one finp'le Letter is afijgned to be in its place, then thofe Quan- tities begin with r, and that when two Letters are affigned to be in their places, they begin with j, and fo on : moreover 'tis ob- vious, that thefe Quantities change their Signs alternately, and that the numerical Coefficients, which are prefixed to them are thofe of a Binomial Power, whofe Index is equal to the number of Letters which are to be difplaced. PROBLEM XXXVI. y^^jy given number of Letters a, b, c, d, e, f, &'c. be- ing each repeated a certain number of times^ and taken promifcuoufy as it happens : To find the Probability that of fome of thofe forts, fo?ne ojte Letter of each may be found i7i its place ^ and at the fame ti7ne^ thatoffojne other forts^ no ofte Letter be found in its place. S L u- The Doctrine £/"Chances. 113 SoLUT ION. Suppofe n be the number of all the Letters, / the number of times that each Letter is repeated, and confequently — the whole number of Sorts : fuppofe alfo that p be the number of Sorts of which fome one Letter is to be found in its place, and q the number of Sorts of which no one Letter is to be found in its place. Let now the prefcriptions given in the preceding Problem be followed in all refpedls, faving that r mufl here be made = - , i-=. , /= -, &c. and the Solution of any n . ft ^— I « . /?— I ft ^"^ 2 ^ particular cafe of the Problem will be obtained. Thus if it were required to find the Probability that no Letter of any fort fhall be in its place, the Probability thereof would be exprefled by the Series 2I1.2 1.2.3 I 1.2.3.4 of which the number of Terms is equal to ^ -|- i. But in this particular cafe q would be equal to y , and therefore, the foregoing Series might be changed into this, viz. I » — / I n — / . n — 2/ . I n — / . n—il. n — 3/ a « — I 6 » — I . n — 2 ~t" 24. » — I . n — 2 . n — 3 » ^^' of which the number of Terms is equal to -^^ . Corollary i. From hence it follows, that the Probability of one or more Letters, indeterminately taken, being in their places, will be ex- prefled as follows : n—l , I n—l . n — 2/ I _^ » — / . n — 7/ . n — %l "2+" I n—l , I n—l . n — 2/ I » — / . n — 7/ . n — %l Co ROLLARY 2. The Probability of two or more Letters indeterminately taken being in their places, will be + -7^ X 737 -i^' &c. wherein it is neceflary to obferve, that the Capitals A^ B, C, D, Sec. denote the preceding Terms. Q^ Altho' 114 ^^ Doctrine of Chaucer. Altho' the formation of this lafl Series flows naturally from what we have already eftabliflied, yet that nothing may be want- ing to clear up this m.atter, it is to be obferved, that if one Species is to have fome one of its Letters in its proper place, and the reft of the Species to be excluded, then the Series whereby the Problem is determined being Jo begin at r, according to the Precepts given in ihe preceding Problem, becomes but then the number of Species being j , and all but one being ft fl •— / to be excluded, it follows that tj in this cafe is = 7— i = — 7— wherefore the preceding Series would become, after the proper Sub- ftitutions, / n — /./ , I »— / . n—zl.l I H — l.n — il . n — 3/./ o^ ~ — -4 X T X > occ. a n.n- — l ' 2 n . n — i.n — 2 6 n.n — l.n — 2.B — 3 And this is the Probability that fome one of the Letters of the Species particularly given, may obtain its place, and the reft of the Species be excluded ; but the number of Species being ~ , it fol- lows that this Series ought to be multiplied by 7 , which will make it » — / , I « — / . n — 2/ I n — / . n — 2/. n — xl . _ ___ ^___ _l ■ y — — X ■ ^ &C •* n — I *^ 2 n — I • « — 2 6 n — i . n-—z . n — 3 ' **'*" And this is the Probability that fome one Species indeterminately taken, and no more than one, may have fome one of its Letters in its proper place. Now if from the Probability of one or more being in their places, be fubtraded the Probability of one and no more being in its place, there will remain the Probability of two or more indeterminately taken being in their places, which confequently will be the difference be- tween the following Series, viz. I n — / j^ I n—l . t i — 2/ ' V. "—l ■ n—zl. n — 3/ « 1 — 7 X "lirr "T ? ^ »-i . »_2 "^ 1; n-i . n-Z . «-3 ' "'*^* , n—l,l n — / . n— 2/ 1 „ "— I ■ n-rzl . a— 3/ « and I — X r X ^ , occ. *"" * n I ' 2 « — I • n — 2 t) n — i . n—z . n — 3 ' which difference therefore will be I 7155 Doctrine (?/" Chances. 115 or-x X A A ^— X B r-rx C.&c. 2 n—\ 1.3 »— 2 I 2.4 n— 3 3.5 n — 4 » as we had expreffed It before : and from the fame way of reafoning, the other following Corollaries may be deduced. Corollary 3. The Probability that three or more Letters indeterminately taken may be in their places, will be expreffed by the Series 1 «— / ■ «— 2/ ^ " %l y, y 4 tt — ^l p ■g" X „_i . »-2 "" I . 4 ^ »— 3 ^"1" 2.5 ^ »— 4 -^ -r X — C-\ -X — -rD., &c. 3.6 » — 5 I 4 . 7 n — & ' Corollary 4. The Probability that four or more Letters indeterminately taken may be in their places will be thus expreffed I „— / . n—ii . f,—ii _i_ V J— i{ ^ J L_ V .Zz£r 24 ^ »— I . »— 2 . «— 3 1.5 ^ n— 4 -^ "1" 2 . 6 ^ a— 5 ^ X r- C, &C. 3 . 7 n — o » The Law of the continuation of thcfe Series being manifeft, it will always be eafy to aflign one that fhall fit any cafe propofed. From what we have faid it follows, that in a common Pack of 52 Cards, the Probability that one of the four Aces may be in the firft place, or one of the four Deuces in the fecond, or one of the four Trays in the third ; or that fome of any other fort may be in its place (making 1 3 different places in all) will be expreffed by the Series exhibited in the firft Corollary. It follows likewife, that if there be two Packs of Cards, and that the order of the Cards in one of the Packs be the Rule whereby to eftimate the rank which the Cards of the fame Suit and Name are to obtain in the other j the Probability that one Card or more in one of the Packs may be found in the fame pofition as the like Card in the other Pack, will be expreffed by the Series belonging to the firft Corollary, making « = 52, and /= i. Which Series will in this cafe be i — r .-7 -f -^ — -^ -I '- 1— , &c. whereof ,2 ' O 1^ ' 120 720 52 Terms are tb be taken. Q2 If ii6 Ite Doctrine ij/'Chances, If the Terms of the foregoing Series arc joined by Couples, the Series will become l-L-J L '—A '- I ' 2 ' 2.4 1 2.3.4.6 I 2.3.4.5.6.8 T^z. 3. 4. 5. 6. 7. 8. 10 &c. of which 26 Terms ought to be taken. But by reafon of the great Convergency of the faid Series, a few of its Terms will give a fufficient approximation, in all cafes -, as. appears by the following Operation — =0.500000 — = 0.125000 3 •4- S -fe. 7 2.3.4.6 =0-006944+- ■^i-^-g-^ =0.000174 + =0.000002 + in ' Sum ==0.632120 + Wherefore the Probability that one or more like Cards in two different Packs may obtain the fame pofition, is very nearly 0.632, and the Odds that this will happen once at lead as 632 to 368, or 1 2 to 7 very near. But the Odds that two or more like Cards in two different Packs will not obtain the fame pofition are very nearly as 736 to 264, or 14 to 5. Remark. It is known that i Ary -\- — y y -\- "^ y^ ~l" TT y^^^- '^^ the num- ber whofe hyperbolic Logarithm is y, and therefore i —^ -^ Lyy _^^3 _j — L_^+ &c. is the Number whofe hyperbolic Lo- garithm is — y. Let A^ be =:/ — ^yy + -^ ;'? j-;'* &c. then I — iV is the Number whofe hyperbolic Logarithm is — y. Let now ^ be = 1, therefore i — iV is the number whofe hyperbolic Logarithm is — 1 ; but the number whofe hyperbolic Logarithm is — 1, is the reciprocal of that whofe hyperbolic Logarithm is i, or whofe Briggian Logarithm is 0.4342944. Therefore 9-5657056 i& the Briggian Logarithm anfwering to the hyperbolic Logarithm — I, but the number anfwering to it is 0.36788. Therefore i — N-= 0.36788 ; and N=^ i — 0.36788 =: 0.63212 j and there- fore 7^5 Doctrine ^ Chances* 117 fore the Series _y — -ijyr^_|-I.^j— ._L.^4 &€. in infinitum^ when y = I, that is i — 1 + 1 — -^ &c. = 0.67212. ^ 2 * 6 24 J Corollary 5. If A and 5 each holding a Pack of Cards, pull them out at the fame time one after another, on condition that every time two like Cards are pulled out, A (hall give B a Guinea ; and it were required to find what confideration B ought to give A to play on thofe Terms : the Anfwer will be one Guinea, let the number of Cards be what it will. Altho' this be a Corollary from the preceding Solutions, yet it may- more eafily be made out thus ; one of the Packs being the Rule whereby to eflimate the order of the Cards in the fecond, the Pro- bability that the two firfl Cards are alike is -^ , the Probability that the two fecond are alike is alfo — , and therefore there be- ing 52 fuch alike combinations, it follows that the Value of the whole is — =1^ COROL L A R Y 6. If the number of Packs be given, the Probability that any given number of Circumftances may happen in any number of Packs, will eafily be found by our Method : thus if the number of Packs be k, the Probability that one Card or more of the fame Suit and Name in every one of the Packs may be in the fame pofition, will. be expreflTed as follows, . n — I I t . n ]k-z > &C. . » — I . »— 2 . n — V n 2 . n . n — 1| 6 . n . n — I . n — i, I 24 . n . » — 1 . »— 2 . n — 3/ PROBLEM XXXVII. If A and B play together each with a certain 7tumber of Bowls = n : what are the refpe&ive Probabilities of winning, fuppofing that each of them wa?it a certai?t number of Games ofbeiftg up ?> SOLU- I J 8 7^^' Doctrine ^Ckances. Solution. Firjl^ The Probability that fome Bowl of B may be nearer the Jack than any Bowl of y^ is — 5 A and B in this Problem being fuppofed equal Players. ^ • _ ISecondlyy Suppoiing one of his Bowls nearer' the Jack than any Bowl of /I, the number of his remaining Bowls is n — i, and the number of all the Bowls remaining between them is 2k — - i : wherefore the Probability that fome other of his Bowls may be nearer the Jack than any Bowl of A will be J~| ■, from whence it follows, that the Probability of his winning two Bowls or more is — X 2 Thirdly^ Suppofing two of his Bowls nearer the Jack than any Bowl of y^, the Probability that fome other of his Bowls may be nearer the Jack than any Bowl of A, will be —Ett ' therefore the Pro- bability of his winning three Bowls or more is ^ x ~Er~ ^ ~zr~ • the continuation of which procefs is manifeft. Fourthly y The Probability that one finglc Bowl of 5fhallbenearer the Jack than any Bowl of ^ is 7 — 7 x -^^ or 7 x J_^ ; for if from the Probability that one or more of his Bowls may be nearer the Jack than any Bowl of A, there be fubtraded the Probability that two or more may be nearer, there remains the pro- bability of one finglc Bowl of J3 being nearer : in this cafe B is faid to win one Fowl at an end. Fifthly, The Probability that two Bowls of B, and not more, may be nearer the Jack than any Bowl of A, will be found to be L X ■ "~' X — - — , in which cafe, B is faid to win two Bowls 2 Z« — I zn—z at an end. Sixthly, The Probability that B may win three Bowls at an end will be found to be 7 x -^^ x -£^ x -^ ; the pro- cefs whereof is manifeft. It may be obferved, that the foregoing Expreflions might be re- duced to fewer Terms ; but leaving them unreduced, the Law of the Procefs is thereby made more confpicuous. It is carefully to be obferved, when we mention henceforth the probability of winning two Bowls, that the Senfe of it ought to be extended to two Bowls or more j and that when we men- tion The Doctrine ©/"Chances. 119 tion the winning two Bowls at an end, it ought to be taken in the common acceptation of two Bowls only : the like being to be obferved in other cafes. This preparation being made ; fuppofe that A wants one Game of being up and B two ; and it be required in that circumftance to determine their probabilities of winning. Let the whole Stake between them be fuppofed = r. Then either A may win a Bowl, or B win one Bowl at an end, or B may win two Bowls. In the firfl: cafe be lofes his Expectation, In the fecond cafe B becomes intitled to ^ of the Stake, but the probability of this cafe is | x -^^ • Wherefore his Expec- tation arifing from that part of the Stake he will b? intitled to, if this Cafe (hould happen, and from the probability of its happening, will be— X - ^^_ ^ ' In the Third cafe, B wins the whole Stake i, but the probability ... - . I n — I of this Cafe, is 7 x ^^_, - . From this it follows, that the whole ExpeQation of B is equal to - X — ^ 1- -L X -^^^^ , or ^^^^ . Which being fubtraded from Unity, the remainder will be the Expecftation of A, viz. ■^—- . It may therefore be concluded that the Probabilities which A and B have of winning are refpeftively as 5« — 2 to 3« — 2. 'Tis remarkable, that the fewer the Bowls are the greater is the proportion of the Odds ; for if A and B play with fingle Bowls, the proportion will be as 3 to 1 j if they play with two Bowls each, the proportion will be as 2 to i j if they play with three Bowls each, the proportion will be as 13 to 65 yet let the number of Bowls be never fo great, that proportion will not defcend fo low as 5 to 3- Let us now fuppofe that A wants one Game of being up, and jB three ; then either A may win a Bowl, or B one Bowl at an end, or two Bowls at an end, or three Bowls. In the fir ft Cafe, B lofes his Expectation. If the fecond Cafe happen, then B will be in the circumftance of wanting but two to A's one j in which cafe his Expectation will be '^^^_ ' , as it has been before determined : but the probability that this*Cafc may happen is 4- x — ^ j wherefore the Expecta- tion I20 7"/^^ Doctrine «/ Chances. tion Off B arifing from the profped of this Cafe will be equal to 1 » T.lt—2 — X X -? 2 2« — 1 s« — 4 If the third Cafe happen, then B will be intitled to one half of the Stake : but the Probability of its happening is — x "~' ■ x — - — ; wherefore the Expe<3:ation of B arifing from the profpedl of this cafe is - x x -— — ■ or -^ 4 in — I 2'? — 2 8 2« — I If the fourth Cafe happen, then B wins the whole Stake i : but the Probability of its happening is 7 x £^7" ^ zn—z °^ I n — 2 — X ^ zn — I From this it follows, that the whole Expeftation of B will be q«i>— 1^» 4_ _ ^j^j(.}^ being fubtradked from Unity, the remainder 8 X zn — 1 !■ will be the Expedation of ^, vtz. — — -^— -.^ . It may there- fore be concluded that the Probabilities which A and B have of winning will be as 23«« — ign-\-/\.to gnn — i3«-l-4. N. B. If A and B play only with one Bowl each, the Expeda- tion of B as deduced from the foregoing Theorem would be found n= o, which we know from other principles ought to be = -|- , The reafon of which is, that the cafe of winning two Bowls at an end, and the cafe of winning three Bowls enter the general con- clufion, which cafes do not belong to the Suppofition of playing with lingle Bowls ; wherefore excluding thofe two Cafes, the Ex- pedation of B will be found to be 7 x ^^_ - x ■• '^,^_^ = j , which will appear if n be made = i. But the Expedation of B in the cafe of two Bowls would be rightly determined from the general Solution : the reafon of which is, that the Probability of winning three Bowls being univerfally - x ^^^_ ~ - , that Expreflion becomes = o, when « is interpreted by 2 ; which makes it that the general Expreflion is applicable to this Cafe. After what has been faid, it will be cafy to extend this way of reafoning to any circumftance of Games wanting between A and B ; by making the Solution of each Ampler Cafe fubfervient to the Solution of that which is next more compounded. Having given formerly the Solution of this Problem, propofed to mc by the Honourable Fraficis Robartes Efq;, in the Pbilofophical TrafjfaSiiom Number 329 j I there faid, by way of Corollary, that if Ihe Doctrine <?/ Chances. 121 if the proportion of Skill in the Gamefters were given, the Problem might alfo be folved : fince which time M. de Monniort, in the fccond Edition of a Book by him publiflied upon the Subjedl of Chance, has folved this Problem as it is extended to the confideration of the Skill, and to carry his Solution to a great number of Cafes, giving alfo a Method whereby it might be carried farther : But altho' his Solution is good, as he has made a right ufe of the Dodrine of Combinations, yet I think mine has a greater degree of Simplicity, it being dedu-« ced from the original Principle whereby I have demonflrated the Dod:rine of Permutations and Combinations : wherefore to make it as familiar as pofTible, and to fhew its vaft extent, I fhall now ap- ply it to the general Solution of this Problem, by taking in the con- fideration of the Skill of the Gameflers. But before I proceed, it is necefiary to define what I call Skill : viz. that it is the proportion of Chances which the Gamefters may be fuppofed to have for winning a fingle Game with one Bowl each. PROBLEM XXXVIII. ' If A and B, whofe proportion of Jkill is as a to b, play together each with a certain number of Bowls : what are their refpeEiive Probabilities of wimiing., f^ppof^g each of them to want a certain number of Gatnes of being up f Solution. F/'r/?, The Chance of B for winning one fingle Bowl being h, and the number of his Bowls being w, it follows that the Sum of all his Chances is nb ; and for the fame reafon, the Sum of all the Chances of A\% na : wherefore the Sum of all the Chances for win- ning one Bowl or more is na-\-7ib ; which for brevity's fake we may call/ From whence it follows, that the Probability which B has of winning one Bowl is — . Secondly, Suppofing one of his Bowls nearer the Jack than any of the Bowls of A, the number of his remaining Chances is n — i x b ; and the number of Chances remaining between them is s — b : wherefore the Probability that fome other of his Bowls may be nearer the Jack than any Bowl of A will be "~ /l' — ; from whence it follo ws th at the Probability of his winning two Bowls or more is nh pi — \ -x. b T ^ f-b • R lhirdh\ 122 The Doctrine <?/" Chances. Thirdly^ Suppofing two of his Bowls nearer the Jack than any of the Bowls of ^, the number of his remaining Chances is n — 2x^1 and the number of Chances remaining between them is j — 2^ ; wherefore the Probability that fome other of his Bowls may be nearer the jack than any Bowl of A will be ■='^t~^{, — . From whence it follows that the Probability of his winning three Bowls or more is —j- x — -r-^ — x • / ^^ — j the continuation of which procefs is manifeft. Fourthly, If from the Probability which B has of winning one Bowl or more, there be fubtra(fted the Probability which he has of winning two or more, there will remain the Probability of his win- ning one Bowl at an end : which therefore will be found to be nb nb n — i x ^ nb i — nb nh an X ' . , - , or — T- X —? — r- , or —r- x Fifthly^ For the fame reafons as above, the Probabili ty wh ich B has of winning two Bowls at an end, will be -^ x "T_^^ — x Stxihiy, And for the fame rcafon likewlfe, the Probability which J3 has of winning three Bowls at an end will be found to be ~ X - ""7 ' I — X - "r^l - X /" , i The continuation of which / /— * _ J~zb /— 3* procefs is manifeft, N. B. The fame Expedlations which denote the Probability of any circumftance of B will denote likewlfe the Probability of the like circumftance of A, only changing ^ into a, and a into l>. Thefe things being premlfed, fuppofe Jir/i, that each wants one Game of being up ; 'tis plain, that the Expedations of A and B are refpedlively -j- and ~ . Let this Expedation of B be cal- led P. Secotidly, Suppofe A wants one Game of being up, and B two, and let the Expedation of B be required : then either A may win a Bowl, or B win one Bowl at an end, or B win two Bowls. If the firft Caie happens, B lofes his Expedation, If the fecond happens, he gets the Expedation P ; but the Pro- bability of this Cafe is -^ x -^ : wherefore the Expedation of B arifing from the poflibility that it may fo happen is -^ x -j-zf X P, If 'T}:)e DocTR I N E <?/ Chances. 123 If the third Cafe happens, he gets the whole Stake 1 ; but the Probability of this Cafe is -^ x -■ " . ~ ; wherefore the Expedation ofSarifing from the Probability of this Cafe is — ;- x ._ - x i. From which it follows, that the whole Expedation of B will be -y X ■ _^ X P -|- "7~ X TZT • Let this Expedlation be cal- led Q. Thir^fyt Suppofe yi to want one Game of being up, and B three : then either B may win one Bowl at an end, in which Cafe he gets the Expedlation Qj or two Bowls at an end, in which Cafe he gets the Expedation P > or three Bowls, in which Cafe he gets the whole Stake I. Wherefore the Expedation of B will be foned to be n — 2y.b ^ f—zb • An infinite number of thefe Theorems may be formed in the fame manner, which may be continued by infpedlion, having well ob- fervcd how each of them is deduced from the preceding. If the number of Bowls were unequal, fo that A had tn Bowls, and By n Bowls j then fuppofing ma -\- nb = i, other Theorems might be found to anfwer that inequality : and if that inequality fhould not be conftant, but vary at pleafure ; other Theorems might alfo be found to anfwer that Variation of inequality, by following the fame way of arguing. And if three or more Gamefters were to play together under any circumftance of Games wanting, and of any given proportion of Skill, their Probabilities of winning might be determined in the fame manner. PROBLEM XXXIX. To find the ExpeSiation of A, when wiih a Die of any given number of Faces^ he undertakes to fling any num- ber of them in any given number of Cafis. SOLU T 10 N. Let p -\- \ he the number of all the Faces in the Die, n the number of Cads, f the number of Faces which he undertakes to fling. R 2 The 124 "^be Doctrine c/ Chances. The number of Chances for the Ace to come up once or more in any number of Cafts «, is /> -J- iV — p" • as has been proved in the Introduction. Let the Deux^ by thought, be expunged out of the Die, and thereby the number of its Faces reduced to />, then the number of Chances for the Ace to come up will at the fame time be reduced tQ /,•! — p — y\" . Let now the Deux be reftored, and the number of Chances for the Ace to come up without the Deux, will be the fame as if the Deux were expunged : But if from the number of Chances for the Ace to come up with or without the Deux, viz. i'xomp -1-0" — P' ^^ fubtraded the num ber of Chances for the Ace to come up without the Deux, •viz.p'' — p — iV> there will re- main the number of Chances for the Ace and the Deux to come up once or more in the given number of Cafts, which number of Chances confcquently will be p -\- i]" — 2/' -{■ p — -"TA". By the fame way of arguing it will be proved, that the number of Chances, for the Ace and Deux to come up without the T'ray^ will be*' — 2 X /» — r\ ' -\- p — 2I " , and confequently that the number of Chances for the Ace, the Deux, and T'ray to come up once or more, will be the di fferenc e between/* -f-i^" — ^P" 4" P — ^^ "> and p^ — 7.Y.P — '\\" 4 - p — 2\ " which therefore will be /> + A " Again, it may be proved that the number of Chances for th e Ace^ the Deux, the T ray, and the S^ uatre to come up is p -\- il " — 4 X/*' -f 6 x/* — A " • — 4 X/* — 2^ » + / — 3^ " ; the continua- tion of which procefs is manifeft. Wherefore if all the Powers /i-|- il", p'> , p — i 1«, p — z\" y p — 3Y', &c. with Signs alternately pofitive and negative be writ- ten in order, and to thofe Powers there be prefixed the refpec- tive Coefficients of a Binomial raifed to the Power /, expreffing the number of F;?ees required to come up ; the Sum of all thofe Terms will be the Num erator of the Expedlation of A, of which the Denominator will hQ p -\- iV. Example i. Let Six be the number of Faces in the Die, and let A under- take in eight Cafts to fling both an Ace and a Deux, without any regard to order : then his Expedlatlon will be gg = -TT — T =- nearly. 168021D 7 •' Exam- T^e Doctrine o/" Chances. 125 Example 2. Let ^ undertake with a common Die to fling all the Faces in 12 Cafts, then his Expedlation will be found to be 6"- — dxij^-f"'?**-!'^ — 20x3"+ 15x2" — 6xi"-f-ixo'* TO nearly. Example 3. If A with a Die of 36 Faces undertake to fling two given Faces in 43 Cafts ; or which is the fame thing, if with two common Dice he undertake in 43 Cafts to fling two Aces at one time, and two Sixes at another time ; his Expedation will be JT^ = T^ nearly. N. B. The parts which compofe thefe Expedlations are eafily ob- tained by the help of a Table of Logarithms. PROBLEM XL. To jind in how ma?ty Trials it will be probable that A with a Die of any given number of Faces fhall throw any propofed number of them. Solution. Let/ -^ I be the number of Faces in the Die, and y the num- ber of Faces which are to be thrown : Divide the Logarithm of — J by the Logarithm of -^^^ , and the Quotient will ex- prefs the number of Trials requiflte to make it as probable that the propofed Faces may be thrown as not. Demon stration. Suppofe Six to be the number of Faces that are to be thrown^ and n the number of Trials , then by what has been demonftrated in the preceding Problem the Expedation oi A will be />+ ■ ''-6x/+ Kv^^''-f20>'7^''-Kv/>"^''.f6x7^"4-7^>. /+.'" Let 126 Ibi Doctrine ^Chances. Let it be fuppofed that the Terms, p-\- i, p, p — i, p — 2, &c. are in geometric Progreflion, (which Suppofition will very little err from the truth, efpecially if the proportion of ;> to i, be not very fmall.) Let now r be written inftead of /'^ ■■ ' -", and then the Expeilation of A will be changed into i — -^ — — — -2i- -L -li 1- A 1- , or I L.\ 6 . But this Ex- r-* r^ r^ r r pedtation of A oaght to be made equal to — , fince by Suppo- fition he h as an eq ual Chance to win or lofe, hence will arife the Equation i ^\ = -^ or '■" = \—-- , from which it I— V' - r 2 ^4-t^ „ ^ may be concluded that n Log. r, or « x Log. — - — = tog, r-7 and cbnfequently that n is equal to the Logarithm of — Anrl tVtp Came' H«*rr divided by the Logarithm of — -^ , And the fame dcmonftration will hold in any other Cafe. Example i. To find in how many Trials A may with equal Chance undertake to throw all the Faces of a common Die. The Logarithm of {—^ = 09621753 -, the Logarithm of 1 — v/ — 2 JAJ- or - = 0.07018 12 : wherefore « = -^^^-^'^^ = 12 -1-. J 5 0.0701812 ' From hence it may be concluded, that in 12 Cads y4 has the worft of the Lay, and in 13 thebeft of it. Example 2. To find in how many Trials A may with equal Chance with a Die of thirty-fix Fac«s undertake to throw fix determinate Faces ; or, in how many Trials he may with a pair of common Dice un- dertake to throw all the Doublets. The Logarithm of ^-y- being 0.9621753. and tlie Logarithm I— v/ — a qC 2+1. or — being 0.0122345 ; it follows that the number ef Cafts requifite to that effeft is -^^^^, or 79 nearly. But The Doctrine ©/"Chances. 127 Bat If it were the Law of the Play, that the Doublets muft be thrown in a given order, and that any Doublet happening to be thrown out of its Turn (hould go for nothing ; then the throwing of the fix Doublets would be like the throwing of the two Aces fix times : to produce which, the number of Cafts requifite would be found by multiplying 35 by 5.668, as appears from the Table annexed to our v'** Problem ; and confcquently would be about 198. N. B, The Fradion — j , may be reduced to another form z f viz. —f — > which will facilitate the taking of its Logarithm. PROBLEM XLI. Suppojing a regular Prifm having a Faces marked i, b Faces marked 11, c Faces marked 11 1, d Faces marked IV, &c. what is the Probability that in a certain number of throws n, fo7ne of the Faces ?narked i will be thrown^ as alfo fome of the Faces marked 11 f Solution. Make a A^b \ c A^ d . Sec. = s, then the Probability required will be expreffed by s" — s — a\ " -J- i — a — i?) " ; the Demonflration _ of which flowing naturally from the Method of arguing employed in the xxxix'*" Problem, there can be no difficulty about it. Ex AMPL E. Suppofe it be required to find the Probability of throwing In S throws the two Chances v and vi, with a pair of common Dice. The number of all the Chances upon two Dice being 36, where- of 4 belong to the Chance v, and 5 to the Chance vi ; it fol- lows that s ought to be interpreted by 36, a by 4, and ^ by 5 : which being done, the Probability required will be expreffed by 36.^ 128 The Doctrine <?/'Chances. ^^^Z^fs"^ ^'^*' which by help of a Table of Log, will be ^ , » found thus : •^ — ^^•^^ 0-38975. -^ — 0.30176, •—- = 0.10012, but I — 0-38975 — 0.30176 -j- 0,10012 = 0.40861, and this being fubtrafted from Unity, there remains 0.59139, and therefore the Oddsagainft throwing v and vi in 8 throws are 59139 to 40861, that is about 1 3 to 9. But if it be required, that fome of the Faces marked i, fome of the Faces marked 11, and fome of the Faces marked iii, be thrown, the Probability of throwing thofe C hances in a given n umber of thro ws n will be exprefTed by j" — j — g ^" A^ s — a-^^ " — j — a — b—c\'' — s S\n -|-i — a— P^" — s — cV -4-J — h-^ " s" And if the Faces marked iv are farther required to be thrown, the Probability of it will be exprefTed by i' — s — a\" A^ i — a — b\'' — s — a — b — A ''-f-'t — ^ — f> — c — ^A" — J — A » -^ s — a — f\ ' — s — a — b — ^"l ' — s — c\ » -|- -f — ^ —-d\ " — s — a — c — (C^" — 5 — d\''-\-i — b — c\'' — s — b — c — /\« -Ys — c — d^ Now the order of the preceding Solutions being manifeft, it will be eafy by bare infpeftlon to continue them as far as there is occafion. PROBLEM XLIL If A ohlhes himfelf in a certain number of throws n with a pair of common Dice not only to throw the Chances V and VI, hut v before vi ; with this refiriBion^ that if he happens to throw vi before v, he does not indeed lofe his wa^er^ but is to proceed as if fiothing had been done^ fiill deduEling fo many throws as have been vain from the number of throws which he had at frfi given kim ; to find the Probability of his winning. SOLU- 1U Doctrine ^Chances. 129 Solution. Let the number of Chances which there are for throwing v be called a, the number of Chances for throwing vi, b ; the number of all the Chances upon two Dice/, and thenumber of throws that A takes = n. This being fuppofed, 1°. \i A throws v the firll throw, of which the Probability is y , he has nothing more to do than to throw vi inn — i times, of which the Probability is i — - — -ztt" > ^"^ therefore the Pro- bability of throwing v the firft tim e, and throwing afterwards vi in n — I times is -7 x i '-—^ — ; — 2°, If A miffes v the firft time, and throws it the fecond, of — a a 7~^7 which the Probability is — -^ x 7- , then he is afterwards to throw VI in n — 2 times, of which the Probability being i — O ,n — 2 it follows that the Probability of miffing v the firft time, throwing it the fecond, and afterwards throwing vi, will be -- x y XI — . fit — 2 3'. l( A mifles v the two firft times, and throws it the third, then he is afterwards to throw vi in n — 3 times, the Probability of all which is •^~^- x j- x i — •^~^_^ - j and fo on. Now all this added together conftitutes two geometric Progreffions, the number of whofe Terms in each is « — i. Wherefore the Sum of the whole will be -^=1 — 73r" ^ -Jn : and if a and b are equal, then the fecond part will be reduced to — n — I X rt X s — tf>— ■ X -^ . J" Now for the application of this to numbers ; a in the Cafe propofed is = 4, <^ = 5, j= 36. Let « be = 12, and the Probability re- quired will be found to be 0.44834, which being fubtradted from unity the remainder will be 0.55166, and therefore the Odds againft A are 55166 to 44834, that is nearly as 2 1 to 17. S But 130 l})e Doctrine 0/ Chances. But if the conditions of the Play were that ^ in 12 times fhould throw both vand vi, and that vi {hould come up before v, the Odds againft A would not be fo great; being only 54038 to 45962, that is nearly as 20 to 17. It would not be difficult after what we have faid, tho' perhaps a little laborious, to extend thefe kinds of Solutions to any number of Chances given. PROBLEM XLIir. Any number of Chances bemg given, to find the Proba- ■ bility of their being 'produced iit a given order, without any limitation of the number of ti?72es in which they are to be produced. S O L U T \ ON. 1°. Let the Chances be a and b^ and let it be required to produce them in the order <?, b. The Probability of producing a before b is -^j , which being fuppofed to have happened, b muft be produced of neceffity ; and therefore the Probability of producing the Chances a and b in the given order a, b, is ^ , ^ . 2°. Let the Chances given be <?, b, c, and let it be required to produce them in the order in which they are written ; then the Pro- bability of producing a before /^ or c is "^^TT '■> ^^^^h being fuppofed, the Probability of producing b before c is by the pre- ceding cafe jT^; after which c muft necefiarily be produced, L and therefore the Probability of this cafe is -^^r^ x -p-^ . 3°, Let the Chances be a, b, c, d, and let it be required to pro- duce them in the order in which they are written ; then the Probabi- lity of producing a before all the reft is , ^^^^^ ; which being fuppofed, the Probability of prodocing b before all the remaining is ^ ^^ J which being fuppofed, the Probability of producing C before d is — '-—; . And therefore the Probability of the whole is . , f . , X l\ . X -^ ; and in the fame manner may thefe Theorems be contlnijed in injmitum. And iM The Doctrine ^Chances. ijr And therefore if it was propofed to find the Probability of throw- ing with a pair of common Dice the Chances iv, v, vi, viii, ix, x before vii ; let the Chances be called refpefli^'ely a, b, c, d, e, j\ and m , then the Probability of throwing them in the order they are writ in will be '^ e __ / But as the order in which they may be thrown is not the thhig particularly required here, except that the Chances m arc to be thrown the kit ; fo it is plain that there will be as many different parts like the preceding as the pofition of the 6 Letters o, b, c, d, e,j, may be varied, which being 720 different ways, it follows, that in order to have a compleat Solution of this Queftion, there mull be 720 different parts like the preceding to be added together. However the Chances iv and x, v and ix, vi and vm being re- fpedlively the fame, thofe 720 might be reduced to 90, v/hich be- ing added together, and the Sum multiplied by 8, we fhould have the Probability required. Still thofe Operations would be laborious, for which reafon it will be fufficient to have an approximation, by fuppofing that all the Chances a, b, c, d, e,f, that is, 3,4, 5, 5, 4, 3 are equal to the mean Chance 4, which will make it that the Probability required will be expreffed by 6b sh &i> ■'-■/• ib h ■ -'- - " X -^T-T — X ',' ' X —7- — X TT — or bb+m ^b-^m i^h+m S^+w zb-rm t-^o 24 20 16 12 8 4" . , ? . 2 . 4 . .4 . 4 . 4 . 1024. -^ - X X -— X X — 30 zi> Z2 18 14 10 I . 3 . 5 . 7 . iT. rj- 'JOi^ ' and therefore the Odds againft throwing the Chances iv, v, vi, viir, IX, X before vii are about 13991 to 1024, or nearly 41 to 3. But the Solution might be made flill niore exaft, if inftead of taking 4 for the mean Chance, we find the feveral Probabilities of throwing all the Chances before vii, and take the fixth part of the Sum for the mean Probability ; thus becaufe the feveral Probabi- lities of throwing, all the Chances before vii are refpedtively — i -i- , -i- , -5- , li- , 1 , the Sum of all which is -^^ , if we divide the whole by 6, the Quotient will be -i^ or i;f) 990 "49 S 2 nearly, 132 *The Doctrine o/" Chances. nearly, and this being fuppofed = ■ 7 - wherein z reprefents the mean Chance, we fhall find 2 = 3 — . And therefore the Pro- bability of throwing all the Chances before vii, will be found to be X -^T— x-^T^x -;7r- X -TT-x — ^=: 0.065708 nearly, which 4.14 3S5 320 267 2ob i4r) -*/ ■' ' being fubtrafled from Unity, the remaining is 0.934292, and there- fore the Odds againfl: throwing all the Chances before vii are 934292 to 65708, that is about 14 — to i. But if it was farther required not only to throw all the Chances before vii, but alfo to do it in a certain number of times affigned, the Problem might eafily be folved by imagining a mean Chance. PROBLEM XLIV. //* A, B, C play together on the following conditioju ; Firji that they Jhall each of them flake i^- Seco?tdly that A and B Jhall begin the Play ; Thirdly ^ that the Lofer fhall yield his place to the third Man, which is conflantly to he ohferved afterwards ; Fourthly, that the Lofer fhall be fined a certai?! Sum p, which is to ferve to increafe the common Stock j Laflly, that he fhall have the whole Sum depofted at firft, and in- creafed by the feveral Fines, who fhall firfl beat the other two fucceffvvely : ^Tis demanded what is the Ad- vantage or Difadvantage of A and B, whom we fup- pofe to begin the Play. Solution. Let BA fignify that B beats A, and AC that A beats C, and fo let always the firfl Letter denote the Winner, and the fecond the Lofer. Let us fuppofe that B beats A the firfl; time ; then let us inquire what the Probability is that the Set {hall be ended in any number of Games, and alfo what is the Probability which each Gamefter has of winning the Set in that given number of Games. FirJi, If the Set be ended in two Games, B muft ncceflarily be the winner, for by Hypothelis he wins the firfl time j which may be expreffed by BA, BC. Secondly;, Tlje Doctrine o/"Chances. 133 Secondly, If the Set be ended in three Games, C muft be the winner, as appears by the following Scheme, viz. BA, CB, CA. thirdly. If the Set be ended in four Games, A muft be the win- ner, as appears by the Scheme BA, CB, AC, AB, Fourthly, If the Set be ended in five Games, B muft be the winner, which is thus expreffed, BA, CB, AC, BA, BC. Fijthly, If the Set be ended in fix Games, C muft be the win- ner, as appears ftill by the following procefs, thus, BA, CB, AC, BA, CB, CA. And this procefs recurring continually in the fame order needs not be profecuted any farther. Now the Probability that the firft Scheme fhall take place is — , in confequence of the Suppofition made that B beats A the firft time ; it being an equal Chance whether B beat C, or C beat B. And the Probability that the fecond Scheme fliall take place is — : for the Probability of C's beating £ is ^ , and that being fuppofed, the Probability of his beating A will alfo be — j where- fore the Probability of C's beating B, and then A, will be -I- x - I 4 * And from the fame confideration, the Probability that the third Scheme {hall take place is ^ : and fo on. Hence it will be eafy to compofe a Table of the Probabilities which B, C, A have of winning the Set in any given number of Games ; and alfo of their Expedtations : which Expectations are the Proba- bilities of winning multiplied by the common Stock depofited at firft„ and increafed fucceffively by the feveral Fines. Table »34 The Doctrine o/" Chances. Table of the Probabilities^ &c. jB C A o zX3+2>> ^xs + s/* 3 I — _— 4 8X3+4/ 5 .6X3+5/' A 3, X3+6/. ' .. i y -^^z^-7p. 8 ..8 X3 + 8/. 9 TO 256 X3+9^ 5..X3+10/. &c. Now the feveral Expedations of B, C, A may be fummed up by the following Lemma. Lemma. -TT- + -77— > &c. in vifimtum^ is equal to -^ + -=^ . Let the Expedations of iJ be divided into two Series, viz. 1 . _L. . _J_ . _JL_ , &c. 2 1 ifi I 128 1^ 1C-4 ' 4.^^-1-4.+ _!L__|-_li^, &c. I 2 ' 16 ' iz8 ' 1024 ' The firft Series conftituting a Geometric Progrefllon continually I 2 decreafing, its Sum by the known Rules will be found to be -7- . The fecond Series may be reduced to the form of the Series in our Lemma, and may be thus exprefled 2 *The DocTRiN E o/" Chances. 135 — X— -1-4-4- ^--1- 4t- ■\- —r- > &c. wherefore dividing the whole by -j , and laying afide the firft term 2, we fliall have the Series -| -] ^ _|- -ii- ^ -!J_ ^ &c. which has the fame b+ form as the Series of the Lemma, and may be compared with it : let therefore « be made =r: 5, d-=. 3, and ^ =j: 8, and the Sum of the Series will be — -I — or -^ ; to this adding; the firft Term 2 which had been laid afide, the new Sum will be -^^i- , and that 49 being multiplied by — whereby it had been divided, the product will be — f, which is the Sum of the fccond Series expreffing the 'Expcftation of B : from whence it may be concluded that all the Expedlations of B contained in both the abovementioned Series will be equal to -7- -\ -p. And by the help of the fame Lemma, it will be found that all the Expedlations of C will be equal to — -]- — -/>• It will be alfo found that all the Expedations of A will be equal to ^ 'V —p- We have hitherto determined the feveral Expedations of the Game- fters upon the Sum by them depofited at firft, and alfo upon the Fines by which the common Stock is increafed : it now remains to eftimate the feveral Rifks of their being fined ; that is to fay, the Sum of the Probabilities of their being fined multiplied by the re- fpedive Values of tlie Fines, Now after the Suppofition made of A\ being beat the firft time by which he is obliged to lay down his Fine p, B and C have an equal Chance of being fined after the fecond Game ; which makes the Rilk of each to be = — />, as appears by the followin? Scheme. BA BA — - or — In like manner, it will be found, that C and A have one Chance in four, for their being fined after the third Game, and confequcntly that the Rifk of each is -^^, according to the following Scheme, BA 136 T'he Doctrine (t/'Chances. BA BA CB or CB AC CA And by the like Procefs, it will be found that the Rifk of E and C after the fourth Game is -^p. Hence it will be eafy to compofe the following Table, which cxprefles the Riiks of each Gamefter. Table ofRiJks. B c A -J IP IP hp \p \p i-j -p 3> ITP b^P i/ xz%P xz^P 2^bP z^bP In the Column belonging to B, if the vacant places were filled up by interpolating the Terms -p, —^p, —j'pi ^c. the Sum of the Rifks of B would compofe one uninterrupted geometric Pro- greflion, the Sum of whofe Terms would be = /» 5 but the Terms interpolated conftitute a geometric Progreflion whofe Sum is = 4/ : wherefore if from p there be fubtraded —p, there will remain —p for the Sum of the Rifks of B. In like manner it will be found that the Sum of the Rifks ofC will be = |/>. And lloe Doctrine (p/'Chances. 137 And the Sum of the Rifks of A, after his being fined the firfl: time, will be -p. Now if from the feveral Expedtations of the Gamefters, there be fubtradled each Man's Stake, and alio the Sum of his Rifles, there will remain the clear Gain or Lofs of each of them. Wherefore, from the Expedlations of i? = 4- + ~—p- Subtradling jirft his Stake = i Then the Sum of his Rifles = —p. There remains the clear Gain of 5 = -i -|- -iL /, , . b 7 I + Likewife from the Expedlations of C Subtrading Jir/i his Stake Then the Sum of his Rifles There remains the clear Gain of C = ^ -4- — *, 7 ' 49-^ In like manner, from the Expedations of^= — -\~ -^. Subtrading, Jirjl his Stake = i Secondly, the Sum of his Rifles = ^p, Laftly, the Fine p due to' the Stock by the Lofs of j the firft Game '!= There remains the clear Gain of A 4 ^ 7 ig-* ' But we had fuppofed, that in the beginning of the Play A was beaten ; whereas A had the fame Chance to beat B, as B had to beat him: wherefore dividing the Sum of the Gains of B and A into two equal parts, each Part will be -^ —p, which confequently muft be reputed to be the Gain of each of them. Corollary i. The Gain of C being -\- —p, let that be made r= o 7 49 then p will be found to be = -^ . If therefore the Fine has the fame proportion to each Man's Stake as 7 has to 6, the Gamefters play all upon equal terms : But if the Fine bears a lefs proportion T to 138 The Doctrine 0/ Chances. to the Stake than 7 to 6, C has the difadvantage : thus fuppofing /> = I , his Lofs would be -^ , but if it bears a greater proportion to the Stake than 7 to 6, C has the advantage. CoROLL ARY 2. If the common Stake were conftant, that is if there were no Fines, then the Probabilities of winning would be proportional to the Ex- pedlations ; wherefore fuppofing /> = o, the Expedtations after the firft Game would be — , — ^ - ■, whereof the firft belongs to 7 7 ' 7 ° By the fecond to C, and the third to A : and therefore dividing the Sum of the Probabilities belonging to B and A into two equal parts, it will follow that the Probabilities of winning would be proportional to the numbers 5, 4, 5, and therefore it is five to two before the Play begins that either A or B win the Set, or five to four that one of them that (hall be fixed upon wins it. Corollary 3. Hence likewife if three Gamefters A, B, C, are engaged in a Pou/e, and have not time to play it out ; but agree to divide (S) the Sum of the Stake and Fines, in proportion to their refpeftive Chances : - S will be the Share of B, whom we fuppofe to have got one Game j - S that of C, who fhould next come in ; and — S the Share of A who was laft beat. For, as they agree to give over playing, all con- fideration of the fubfequcnt Fines p is now fet afide, and the Cafe comes to that of the fiift part of Corol. 2. Or the fame thing may be fhortly demonftrated as follows. Put 5=1, and the Share o^ A =^ z. Then B playing with C has an equal Chance for the whole Stake S, and for being reduced to the prefent Expedlation of A; that is, B's Exped-ation is -y". C has an equal chance for o, and for JS's prefent Expedlation; that is, C's Expedation is = — — — . But the Sum of the three Expedations z -\- - X 1 -{• z -\-- xi-\-z = Sz=.ii or z -\- - z f= —z)=z— : and z = ~ , which is As Share ; thofe '4 I 4 -/ ± 7 of B and C being -^ X I + - > and -xi4"""iOr~ ^"^ f > ^^' fpedively. PRO* 'the Doctrine <?/" Chances. 39 PROBLEM XLV. If four Gameflers play on the conditions of the foregoing Problem^ and he be to he reputed the Winner who beats the other three fucceffively., what is the Advantage of A and B whom wefuppofe to begin the Play f Solution. Let BA denote as in the preceding Problem that B beats A, and AC that A beats C ; and univerfally, let the firft Letter always de- note the Winner, and the fecond the Lofer. Let it be alfo fuppofed that B beats A the firft time : then let it be inquired what is the Probability that the Play fhall be ended in any number of Games j as alfo what is the Probability which each Gamefter has of winning the Set in that given number of Games. Firft, If the Set be ended in three Games, B muft neceflarily be the Winner ; fince by hypothefis he beats A the firft Game, which is exprefled as follows : BA 2 3 BC BD Secondly, If the Set be ended in four Games, C muft be the winner ; as it thus appears. BA 2 3 4 CB CD CA Thirdly, If the Set be ended in five Games, D will be the Win- ner ; for which he has two Chances, as it appears by the following Scheme. BA BA BC I 2 3 4 5 CB DC or DB DA DA DB DC T 2 Fourthly, CB CB BC DC CD DB AD AC AD AB AB AC AC AD AB 1 40 T'he Doctrine o/" Chances. Fourthly^ If the Set be ended in fix Games, A will be the Win- ner } and he has three Chances for it, which are thus coUedted. 1 BA BA BA 2 3 4 5 6 Fifthly, If the Set be ended in feven Games, then B will have three Chances to be the Winner, and C will have two, thus ; BA BA BA BA BA CB CB CB BC BC DC DC CD DB DB AD DA AC AD DA BA BD BA CA CD BC BC BD CB CB BD BA BC CD CA Sixthly, If the Set be divided in eight Games, then D will have two Chances to be the Winner, C will have three, and B alfb three, thus ; I BA BA BA BA BA BA BA BA CB CB CB CB CB BC BC BC DC DC DC CD CD DB DB DB AD AD DA AC AC AD AD DA BA AB BD BA AB CA AC CD CB CA CB DB DA BC BA BC CD CD CA DC DC BD BD BA CA CB CD DA DB BA BC BD 3 4 5 6 7 2 3 4 5 6 7 8 Let now the Letters by which the Winners are denoted be written in order, prefixing to them the numbers which exprefs their feveral Chances for winning j in this manner. 3 4 5 6 7 8 9 10 2cc. iB iC 2D 3A 3B4-2C 3C4-2D-W3B 3D-I-2A-I-3C -I-3D + 2A 3A + 2B + 3D 4- 3A + 2B + 2C + 3A + 3D ThcQ 'The Doctrine ^Chances. 141 Then carrying this Table a littler farther, and examining the Forma- tion of thefe Letters, it will appear ; Firjl^ that the Letter B is always found fo many times in any Rank, as the Letter A is found in the two preceding Ranks : Secondly, that C is found fo many times in any Rank as B is found in the preceding Rank, and D in the Rank before that. Thirdly, that D is found fo many times in any Rank, as C is found in the preceding, and B in the Rank before that : And, Fourthly, that A is found fo many times in any Rank as D is found in the preceding Rank, and C in the Rank before that. From all which it may be concluded, that the Probability which the Gamefter B has of winning the Set in any number of Games, is -^ of the Probability which A has of winning it one Game fooner, together with -^ of the Probability which A has of winning it two Games fooner. The Probability which C has of winning the Set in any given number of Games, is - of the Probability which B has of win- ning it one Game fooner, together with - of the Probability which 4 D has of winning it two Games fooner. The Probability which D has of winning the Set in any num- ber of Gam.es is - the Probability which C has of winning it one Game fooner, and alfo - of the Probability which B has of win- ning it two Games fooner. The Probability which A has of winning the Set in any num- ber of Games is - of the Probability which D has of winning it one Game fooner, and alfo - of the Probability which C has of winning it two Games fooner. Thefe things being obferved, it will be eafy to compofe a Table of the Probabilities which B, C, D, A have of winning the Set in any number of Games, as alfo of their Expedations, which will be as follows : I42 The Doctrine o/" Chances. B C \ D \ y^ I II III IV V VI VII Vlll IX X &c. 3 4 5 6 7 8 9 lO 1 1 12 &C 7x4+3/ I — ___ JX4+ 4p * i 1 ,6x4+ 5P -^X4 4- 6/ ■ix4-l- 7^ .Z8X4+ 8/ ' v,-n — :' ' 64 ^4 + 7/ "128x44- 8/ Tj6'X4-f 9/ 1 1 .z8^4+ Sp 6 — ' .50^44- 9p -256^44- 9/ 5,,x44-io> 5.,X4-flO/ ] 5..>^4 + 'cp '.02+x44-ii/._^,^_^X4 + ii/. .0^^44-11/ .04^^44-12/ >^o,sX4 + 12/> ^^^3X4 + 12/. 48^44-12/ The Terms whereof each Column of this Table is compofed, being not eafily fummable by any of the known Methods, it will be convenient, in order to find their Sums to ufe the following Let B' -1- B'' + B " 4- B"" -\- B^ -f B^', &c. rcprefent the refpec- tive Probabilities which B has of winning the Set, in any number of Games anfwering to 3,4, 5, 6, 7, 8, &c. and let the Sum of thefe Probabilities in infinitum be fuppofed =_y. In the fame manner, let C -[- C" -j-C'' + C^ 4- C*' + C^', &c. reprefent the Probabilities which C has of winning, which fuppofe Let the Probabilities which D has of winning be reprcfented by ly _j_ D" + D'" -Y D'''-f D^ -1- D^', &c. which fuppofe = v. Laftly, Let the Probabilities which j4 has of winning be reprc- fented by A' -1- A" + A'" + A'^+ A^4- A^'^ &c. which fuppofe Now from the Obfervations fet down before in the Table of Pro- babilities, it will follow, that B' T^tf Doctrine ^ Chances. 143 B'=B' B" = W B'"=-A" + -A' B^=-1.A'^+ lA'* ' 4 B'''=-A''4--iA'*' 2 4 &c. From which Scheme we may deduce the Equation following, y = — 4- —x: for the Sum of the Terms in the firft Column is equal to the Sum of the Terms in the other two. But the Sum of the Terms in the firft Column h ■=:: y by Hypothefis; where- fore y ought to be made equal to the Sum of the Terms in the other two Columns. In order to find the Sum of the Terms of the fecond Column, I argue thus, A' + A'' + A''''-f- A'^+ ^y + A^', &c. r= ;c by Hytoth. Theref. A" + A'"+ A'^+ AT + AT', &c. = x — A' and -A"-|- J.A'"+-A"'+-A^+- A^&c. = -x— -A' 2 '2 '2 '2 '2 2 2 Then adding B' ■\- B" on both Sides of the laft Equation, we (hall have B' + B" +7A" +-^A'"4- -iA'^+^ A^-f jA^ &c. = Lx— 4A'4-B'+B". But A' = o, B' = ~ , B" = o, as appears from the Table : wherefore the Sum of the Terms of the fecond Column is = -ar 2 The Sum of the Terms of the third Column is - x by Hypo- thefis ; and confequently the Sum of the Terms in the fecond and third Columns is = 4 -"^ + ~" . ^om whence it follows that the 4 4 Equation yz=z — -\-lx had been rightly determined. ^ And 144 ^^ Doctrine o/" Chances. And by a reafoning like the preceding, we fliall find z r=.^y -|- -"y, and alfo v=. ^z -\- -y, and laftly x^=—'v -\- —z. Now thefe four Equations being refolved, it will be found that B' -I- B" 4- B'" -\- B'^+ B^ + B^ &c. = ;' = -^ C' 4- C" -\- C" + C'^ + C^+ e-', &c. = 2; = -^ D' -I- D" -y D"^4- D'^ + D^ 4- D^, &c. = -u == -^ A' + A" + A"' + A'^ + A'' -\- A""', Scc. = x= -^ Hitherto we have determined the Probabilities of winning: but in order to find the feveral Expectations of the Gamefters, each Term of the Series expreffing thofe Probabilities is to be multiplied by the refpedive Terms of the following Series, 4 -1- 3/>, 4 + 4/. 4 + 5/-. 4 + ^Z". 4 + Z/*. 4 + §/». &c. The firft part of each produ6l being no more than a Multiplica- tion by 4, the Sums of all the firft parts of thofe Produdls are only the Sums of the Probabilities multiplied by 4 ; and confequently are 224 144. 128 100 r n.- 1 4v, 4^, 4'u. 4x, or -7^, ^^, -7;^, -y^, refpedbvely. But to find the Sums of the other parts, Let 3B> -f 4B> + jB''/. + 6B">, &c. be = pt. 2Cp -\-4C'p -i- 5C"p-\- 6C7, &c. be =ps. 2D'p-\- 4.D"p-\- 5D'"p-\- 6D'7, &c. be =/r. 3A'/.4- 4A''/> -1- 5A''p-\- 6A''7>, &c. be — /^. Now fince 3B''=3B' 4B" = 4B'' sB'"=={a" +^A' 6B'^= -A'" -1- -A" 8B^'=-A^ -4- -A''' 2 ' 4 it follows that t=^ -[- ^q-\-x-> for 1°, the firfl Column Is = / by Hypcthejis. 2°, 3 A' -\- 4A'' -1- 5A''''+ 6A'^ 4- 7A^, &c. = qhy Hypothefi. 3°, A' + A'' + A'"+ A'^ -f A*-, &c. has been found = -^ = to the value of x. Where Ihe Doctrine ^Chances. 145 Wherefore adding thefe two Equations together, we fhall have 4A' + 5A" + 6 A"' + 7A'-^ -f 8A^ &:c. = ^ + x, or fA' + -^A'^ + 4a'"+ \M^\ |a-. &c. = ^? + 7:^-. But A' =: o, therefore there remains ftill -LA'/ + 4 A'" 4- -lA-'+ iA-, &c. r= -^^ + 7^. Now the Terms of this lad Series, together with 3B' + 46", compofe the fecond Column : but 36'=— and 4B" = o, as appears from the Table j confequently the Sum of the Terms of the fecond Column is = -^ + 7? 4" 7^- By the fame Method of proceeding, it will be found that the Sum of the Terms of the third Column is = — y A- —^x. From whence it follows, that^r=— A^ —qA^ —x ■\- o'^^ = 7 + f? + ^- And by the fame way of reafoning, we (hall find i = 7? + I;' -|- -^r + j-u, and alfo r—'-s^^z-\--^t-\- \y, and laftly I I I t I I I ?=7'' + 7'" + 7'+7^- But for avoiding confufion, it will be proper to reftore the va- lues of *•, y, Zj V, which being done, the Equations will fland as follows. 4 n^ 4J I 149 "' •• 596 I 4-' 5= 1 -f- —r. '49 ' 2 ' 4 149 ' 2 '4 2 J49 ' 2 » 4 Now the foregoing Equations being folved, it will be found that 22201 ' ZZlOl ' 22201 ' " 22201 * From which we may conclude that the feveral Expedations of By C, D, ^, &c. are refpedively, U Fir/I, 146 The Doctrine o/" Chances. Firfi, 4X^ + ^^> Secondly, 4X^+ ^^. Thirdly, 4 X ^ + ^t : Fourthly, 4 X ^ + ^ff^p. The Expedations of the Gamefters being found, it will be ne ceflary to find the Rilks of their being fined, or otherwife what Sum each of them ought juftly to give to have their Fines infured. In or- der to which, let us form fo many Schemes as are fufficient to find the Law of their Procefs. And Fir/l, we may obferve, that upon the Suppofition of B beat- ing A the firfl Game, in confequence of which A is to be fined, B and C have one Chance each for being fined the fecond Game, as it thus appears : 1 I BA BA 2 I CB BC Secondly, that C has one Chance in four for being fined the third Game, B one Chance likewife, and D two ; according to the fol- lowing Scheme. I 2 3 BA BA BA BA CB CB BC BC DC CD DB BD thirdly, that D has two Chances in eight for being fined the fourth Game, that A has three, and C one according to the follow- ing Scheme. I 2 3 4 BA BA BA BA BA BA CB CB CB CB BC BC DC DC CD CD DB DB AD DA AC CA AD DA A^. B. The two Combinations BA, BC, BD, AB, and BA, BC, BD, BA, are omitted in this Scheme as being fuperfluous ; their difpofition fhewing that the Set mufl have been ended in three Games, and confequently not affeding the Gamefters as to the Probability of their being fined the fourth Game ; yet the number of all the Chances muft be reckoned as being eight ; fince the Probability of any one Circumflance is but g- . Thefe Schemes being continued, it will eafily be perceived that the circumftances under which the Gameflers find themfelves, in refpcdl of their Rifks of being fined, fland related to one another in the fame manner as did their Probabilities of winning; from which iTje Doctrine o/" Chances. 147 which confideration a Table of the Ri/ks may eafily be compofed as follows. A Table ofRiJks. 2 B c D A / -J -,f // 3 hP 7t 7f /// 4 if \P \P IV 5 -^P i:P i:P -hP V 6 6 fJ ,^P ilP VI 7 ^/ 'p 8 64/ 64/ Vll 8 ^/ iz^P XZSP -ri? Vlll &c. 9 '7 , '' P 256^ Wherefore fuppofing B -|- B" + B'^', &c, C + C^ -|- C", &c. D/ 4. D" -f D'", &c. A' + A"-l- A'". &c. toreprefent the feveral Probabilities ; and fuppofing that the feveral Sums of thefe Probabi- lities are refpedlively y, x, z, v, we fhall have the following Equa- tions X = -V -\- -z. Which Equations being folved we (hall have^ = > •'' - — 224 '75 Xr= Let now every one of thofe Fractions be multiplied by/', and the Produas -^/., -^, -^p, -^p will exprefs the refpedive Rifles of B, C, D, j4, or the Sums they might juftly give to have their Fines infured. But if from the feveral Expe6latIons of the Gamefters there be fub- tradted, Fir/l, the Sums by them depofited in the beginning of the Play, and Secondly, the Rifles of their Fines, there will remain the clear Gain or Lofs of each. Wherefore U 2 From 148 T'he Doctrine ij/'Chances. From the Expedlations of 5 = -^^ 4- -ii^d. _ 14.9 22201 •' Subtradling his own Stake = i And alfo the Sum of his Rifks r^ -^^6. ' "j f There remains his clear Gain = -21 1 ili2_ >, 140 ' 22201 r* From the Expeftations of C= -^^ -I — ^^-^*. ^ 149 ' 22201 -f Subtracting his own Stake = i And alfo the Sum of his Rifks = "TT^P' '49 There remains his clear Gain = T^"^ TH^f- 128 , 37600 From the Expedlations of D = ~^ j- -j^p. Subtradling his own Stake = i And alfo the Sum of his Rifks := ~T7^P' There remains his clear Gain = ~ -[- -^^/. From the Expedations of ^ = -7^ A^ 22101 P Subtracting his own Stake = i And alfo the Sum of his Rifks = Laflly, the Fine due to the t Stock by the Lofs of the i = firfl Game J There remains his clear Gain= rV"^ llzo? P' The foregoing Calculation being made upon the Suppolition of JB beating A in the beginning of the Play, which Suppofition nei- ther affcfts C nor D, it follows thai the Sum of the Gains between B and A ought to be divided equally ; then their feveral Gains will ftand as follows : Gain The Doctrine ©/"Chances. 149 :jj^ If -1^ ^^^^Z*. which is the Gain oi A ox B be made = o > 149 2. 201 ^ ' then p will be found r= -i^ll j from which it follows, that if ' 2700 each Man's Stake be to the Fine in the proportion of 2700 to 1937, then A and B are in this cafe neither Winners nor Lofers ; but C wins — ^ which D lofes. 225 And in the like manner may be found what the proportion be- tween the Stake and the Fine ought to be, to make C or D play without advantage or difadvantagc ; and alfo what this proportion ought to be, to make them play with any advantage or difadvantagc given. Corollary i. A Spedlator S might at firft, in confideration of the Sum 4 -j- yp paid him in hand, undertake to furnidi the four Gamefters with.- Stakes, and pay all their Fines. Corollary 2. If the Stock is confiderably increafed, and the Gamefters agree either to pay no more Fines, or to give over playing, then 1°. If we fuppofe B to have got the laft Game, by beating out Ay and call the Stock Unity -, the Expedations, or Shares, belong- ing to B, C, D, A, refpeaively, will be -^ , -f^ , -^ ,. '49 2°. If B has got 2 Games, by beating D and A fucceflively, the Shares of B, C, D, A, are -^, -^, -^, -^ • For 5 has now an equal Chance for the whole Stake, or for the loweft Chance of the former Cafe : that is, his Expeftation is worth I 150 The Doctrine of Chances. — X 1 H ^^^ = — ^ . C has an equal Chance for o, and for 2 ' 149 149 i- ' ~ — ; that isj his Expedation is —^ , and in the fame way the Numerators of the Expedations of £) and ^are found. Corollary 3. If the proportion of Skill between the Gamefters be given, then their Gain or Lofs may be determined by the Method ufed in this and the preceding I'roblem. Corollary 4. If there be never fo many Gamefters playing on the conditions of this Problem, and the proportion of Skill between them all be fup- poied to be equal, then the Probabilities of winning or of being fined may be determined as follows. Let W, ly, D', E^, F^, A^, denote the Probabilities which B, C, D, Ey F, A have of winning the Set, or of being fined in any number of Games ; and let the Probabilities of winning or of being fined in any number of Games lefs^by one than the preceding, be de- noted by F, C', D^', E^, F, A^': and fo onj then I fay that B' — 7A" '1- tA"' + ?A'^ + -T^A*- C" = -,B^ -f ^F^ + JE^ + -T^D^ 5^ = 7C^ + 7B^ + T^ + -^^ Now the Law of thefe relations being vifible, it will be eafy to ex- tend it to any other number of Gamefters. Corollary 5. If there be feveral Series fo related to one another, that each Term of one Series may have a certain given proportion to fome one affigned Term in each of the other Series, and that the order of thefe propor- tions be conftant and uniform, then will all thofe Series be exaflly fummable. Remark 7^^ Doctrine of Chances. 15^ Number of Games won by B K— 5 ^\c D E F b c d e f «-4 « i n — 2 « — I 11 I II III IV V B C D E F A b ' c' d' e' f a' b" c" d' e" f a" b" c'" d" e"' f" a" b" c" S"" e" f a"" I Remark. As the Application of the Dodrine contained in thefe Solutions and Corollaries may appear difficult when the Gamefters are many, and when it is required to put an end to the play by a fair diftribu- tion of the money in the Foide ; which I look upon as the moft ufeful Queftion concerning this Game : 1 fhall explain this Subjeft a little more particularly. I. Let us then Suppofe any number of Gamefters, « -(- i (as, in our Scheme, 6) and having written down fo many Letters as there are Gamefters in the Order they are to fucceed one another, place under them their refpecflive fmall Letters, to denote the Probabilities which the feveral Game- fters have of winning the Poide, immediately after their Order of Succeffion is fixt, and before the play is begun. Where note that the Letter b fignihes ambiguoufly the Expectation oi A or of 5; and this Cafe being particular, not to occur again in the fame Potde, may be fepa- rated from the others by a line. We fhall always fuppofc B to be the Winner of the firft Game ; and that A takes the lowed place in the fecond Row of Capitals. Under thefe repeat « — -i Rows of the fmall Letters which, with the fmall flrokes or dots affixed to them mark the Expedtations of the feveral Ganaeflers, when any one Gamefter has got as many Games as is the Number of dots, or that which is marked in Roman Charadlers to the right of the Row. For it is to be obferved, that, after the firfl: Game, the fmall Letters thus marked do not, unlefs by accident, fignify the Expeftations of the particular Gamefters at firfl denoted by their Capitals ; but the Expedtations which belong to the Rank and Column where any Letter flands. For Example, b' docs not denote the Expedtations of him who was fuppofed to get the firft Game, unlefs perhaps he has got two more fucceffivelyj but indefi- nitely, thofe of whatever Gamefler has got 3 Games following. And the other Letters of the fame Row, as c'", d'''^ e"' , fignify the fimultancous Expedtations of the three Gameflers that follow him in the Order of playing. 2. This 152 T'he Doctrine o/" Chances. 2. This preparation being made, it will be obvious in what manner the Expedlations are varied by the Event of every Game ; and how they are always reducible to known Numbers. For if we fuppofe B, the Gamefter who is in Play, to have got 3 Games, for inftance, and to want two more of the Poide ; then his prefent Expeftation being b''^ ^ if he wins the next Game which he is to play with C, the Confequence will be ; 1°. His own Expedlations will be changed into b^^ ; having now got 4 Games. 2°. All the other Expectations in the fame Row, will likewife be transferred to the next inferior (IV.) but marked each by the preceding Letter of the Alphabet : that is, d'"- becomes c^, t"'" becomes d"^, &c. ex- cepting only c'", the Expedation of him who loft the Game, which is thereby reduced to the loweft Expectation a"'. And if B had al- ready gained {n — 1^=) 4 Games, and confequently wanted but one; if he gains this, all the Expeftations c'^, d^'\ e'^, &c. will vanifh together, while b^^ becomes ^=1, the Exponent of Certainty, But if B lofes his Game with C, all the Expectations, of what- ever Rank, are transferred to the Rank I, and their Ratios are reftored as when the firft Game was won : only the Letters are changed into the next preceding. As W^ becomes a\ c'" becomes b'^ d'" becomes £•', and fo on. '^. Now there being fuppofed an equal Chance of B's winning and lofing a Game ; any Expectation of his, as when he has got 3 Games, will be thus expreffcd ; b'" = . "^" ■ ; in which, fubfti- tuting for b''' its equal in our Example ■■ '"'"'' , we (hall have b'" ^= -Itiilll . The fmie way, b'^ =. -i^ = -^^ ; and b' = ''^'J" -. In general; when the number of Games that B wants of gaining the Poule is /;/, then fliall — j— ^ — — be the value of his Expectations. 4. The other Expectations are collected nearly in the fame man- ner. As c"' = "*"' - , in which fubftituting for a"" its equal (in our example) -^JtZ- , we have c'" = ^^^±^ — :^ b' ^ ^ f. The fame way, c" = ^^ b' -\- ^ f -]- ~ e': and f'= ^ b' ■\- ^f -}- ■L e' 4- \ d' ; the number of Terms added to the Games won be- ing always = », and the Letter a' always omitted. From 'The Doctrine ^Chances. 153 From all which it appears, that the Expe(5lation of a Gamefter, in any State of the Play, is exprelled by the Expeclations d, b' , c', &c. after one Game is won : and that thefe, therefore, are firft to be computed. 5. In order to which, I fay, that the Letters b, c, d, c, &c. ex- prefllng, as above, the Chances of the Gamefters, B, C, D, E, &:c. immediately after their Order of playing is fixt by lot, or otherwife ; thefe Chances are in the geometrical ProgrcfTion of i -f- 2 " to 2 ". For either of the Gamefters (as yl) who play the firfl: Game, has i out of 2" Chances of beating all his Adverfaries in one Round. And therefore he may, in confideration of the Sum — ^ x b-\-c-\-(i-{-e 2 give up his expedations arifing from the Probability of that Event, and take the loweft place with the Expedlation e ; the Gamefter Cfucceeding to his place, D to that of C; and fo on. But B hav- ing, on the fcore of Priority, the fame demand upon yf, as A has upon B ; that is, neither having any demand upon the other, the Term —^ x ^ is to be cancelled ; and the Value of A's place, with 2 refpeft to the other Gamefters, reduced to — — x c -i — X d -\~ z" 2 &c. And now each of the Gamefters C, D, E. &c. being raifed to the next higher Expedlation b, c, d, Sec, for which he has paid — ^ of his former Expeftation j it follows that b ^=i i -\ — x r, 2" 2 c = I -| — X dy &c. and that, before the play is begun, every Expeftation is to the next below it as i -|- -7- to i, orasi-J-2'' to 2". Which coincides with Theor. I. of Mr. Nicolas BernoidU in Phil, Tranf N. 341. Thus if the Gamefters are 3, {A) B, C ; their firft Expeiflatlons 2re (5) 5, 4, with the common Denominator 14. If they are 4, {A)B, Cyh, their Expeftations are (81) 81, 72, 64, with the Deno- minator 298. If there are 5 Gamefters, their Expedations are (17^) 17^ 17" X 16, 17 X 16% 163, with their Sum for a Denominator ; that is, (4913), 4913, 4624, 4352, 4096, with the Denominator 22898. And the like for any number of Gamefters. 6, It is plain likewife that the Expedations of all the Gamefters, excepting A and £, remain the fame after one Game is plaid, as they were at firftj c' = c, d'=.d, e'^=ie, &c. becaufe the conteil X in 15+ TT^& Doctrine <?/" Chances. in the firft Game concerns"^ and B alone; its Event making no alteration in the Expeflations of the others : but only railing B's firrt expedation, which was b^ to the Value b' , and diminilhing the equal Expedtation of A by the fame quantity : fo that a} -\- b' = 2b. And therefore, to find all the ExpecT:ations after the firft Game is played, we have now only to compute the firft and lafl: of that Rank, b' and a'. But it was found already that if ;« reprefents the number of Games tliat the laft Winner B wants to gain the Poule, his Expectations in that Circumftance will be equal to ■■ ^ ~'^'' .. . From which, putting 2 m ^=zn — I , which is the Cafe when B has got one Game, and the Expedation b' ; and fubftituting for b' its equal 2b — a' y we fhall get a' As when there are 3 Gameflers, n=2, b=.— and a' = 20 ~~' =-^ = ^. Andb^ = 2b-a'=-^--^^±. 3 42 7 14 14 7 If there are 4 Gameflers, «:^3, (^= -;-j-j and therefore ^'= agi) 298 ^7 298 '^ 7 298 149 149 149 '49 If there are 5 Gameflers, 72=4, b =z "7^^ J whence a'=z , 4013 _i ;«:-'° _I_ — _lZii_ i^':? ^ ^ 22898 ' ^ IS 22898 ^15 22898 11449 • AnAb'=2b-a' = -^ L!il_^_^. SothattheEx- 11449 11 449 "449 pedlations of the Gameflers, B having got one Game, will ftand thus : B C D E A b^ c' d' e' a'. 3056, 2312, 2176, 2048, 1857; thefe numbers exprefTing the Ratios of the Expedations ; and with the Denominator 11449 fub- fcribed, their abfolute quantity ; or the Shares of the whole Stake due to each Gamefter, if they were to give over playing. 7. And thus the Probabilities which the feveral Gameflers have of gaining the Poule may in all Cafes be computed, and difpofed into Tables. But the 6 following, will, 'tis thought, be more than fuffi- cient for any Cafe that happens in play. Table The Doctrine ^Chances. Table I. For a Touk of Th^ee' 155 Games won o A B d Denom. 5+5-1-4=14. I B\ 4! C 7 A- Denom. 7. I, II Tab. II. For a Foule of Four. Denom. 298. BS 81 C 72 B 87 C 36 28 D 64 32 18 25 16 Denom, 149. Tab. III. For a Poule oi' Five. A B o| 491^ I. 11 III. Tab. W. For a Poule of Six I. II III IV. B 3056 4255 6653 c 4624 D 4352 E 4096 Denom, 22898. c 2312 2176 E 2048 . '1666 1088 A • . -1857 1568 1024 2040 1528 1920 • . -1 156 Denom. 11449. 1185921 c 149984 D 1115136 E 108 1 -^44 F 1048576 Denom. 6766882. B 682976 863007 1223069 C 574992 540144 472560 '943193I 34H88 D E 540672 \ 458240: 42253^ 2874961 278784 507904: 481602 409728 A 524288- 50294- 46700S 3973^2 270^36 262144 DO CO o a J Q X 2 One 156 TTdc Doctrine <?/* Chances. One Example will fiiew the Ufe of the Tables : Suppofe 5 Game- fters engaged in a Pouky with this condition, that if it is not ended when a certain number of Games are played, they fhall give over, and divide the Money in proportion to the Chances they fliall then have of winning the Poule. That number of Games being played, fuppofe the Poule rifen to 30 Guineas, and that a Gamefter (5) has got two Games : ^i. how the 30 Guineas are to be fliared ? Divide 3 1 — /, into Shares proportional to the numbers 4255, 2040, &c. (in Tab, III.) which ftand in the Row of Games won II. and thofe Shares will be as follow: 31 4 ^- X '''''' = the Share of /. s. d. 5 II 14 2 And the fame way thofe of C, 6cc. will be C 5 12 3 jD 5 5 8 £ 4 11 8 ^463 L. 31 10 o Note, the pricked Line which is drawn in each of the Tables fe- parates the Chances of the Gamefters who are neceJJ'arily to come into the play before the Poule is won, from the Chances of thofe who may poffibly not come in again ; which lie below that line. And, fetting afide the Column B, all the Chances in any Row above the line are in the continued Ra tio of i -|- 2" to 2". As in Tab. III. d"=—x c", on 920 = I ^ X 2040. The fame is true of the Terms of any Row that lie both below the line. But if one lies above and the other below it, their Relation is different, and is to be found by Art. 3. of this Remark, It remains to compute the Profit and Lofs upon the Fines p : as follows. I. The prefent Expe<flations of a Gamefter who is entering, or to enter, into play, that he ftiall be the Winner, are made up of his feveral prefent Expedations, upon the Events of his coming in once^ twice, tbrice, &c. as is manifeft. And as, immediately after the Order of playing is fixt, it was fhewn that thofe total Expedtations are in the geometrical Progreffion of i -|- 2"to 2", the number of Gamefters being n -\- i -^ io, m any other State of the Peuky their Ratio is always given. But The Doctrine of Chances.. 157 But every time that a Gamefter enters, his Chance of winning in that Turn, is to his Chance of paying a Fine^ as i to 2" — i : and therefore, componendo^ the Sum of a Gamefter's feveral ExpecSlations oi "winning, is to the Sum of his feveral Rifks of paying a Fiyie, in the fame Ratio ; the whole Stake, and ahb each Fine p, being put =: I . And the whole Rifis of the feveral Gamefters are in the fame Ratios as their ExpeBations. Thus in the Cafe of Three Gamefters, whofe Expedlations are -^ , -^- , -^ , their Chances of paying the Fine/> will be the lame Fradtions multiplied into 3 (^= 2" — ■ i) ; that is, they will be -^ , '4 14.' '4 * And the firft Expedations of Fcz/r Gameflers being 81, 81, 72* 64, to the Denominator 298 ; their Chances of being Fined will be the fame Numerators multiplied into 7 (=2" — 1), that is, <;67 ?6-' t;o.t 4aS ^ „. , 1^ ' -liT ' l^T ' -7^ ' refpcdively. Hence again it appears, that the Total of the Fines, or the Sum for which they may be furniflied throughout the Peule, is 2"^^^i "Xp. For the Sum of the Expedations upon the Stake i, is i j and thefe are to the Number of Fines as i to 2" — i. 2. Suppofc now that one of the firft Players of Tbree, as ^, is beat out, and his Fine paid, as mufl always neceflarily happen ; and thence, the Expedlations of getting the Pou/e reduced to c B J - - - : then the Rifles of C and ^ will be 4 , - , refpedlively : whofe Sum — taken from 2 (=2" — 2) leaves ^ for the Fines ofB. In like manner, the Expedlations of Four Gamefters, after one Game is won, being 56, 36, 32, 25, with the Denominator 149 ; the Numerators of the RiiTcs of the Three lafl Gamefters C, D, /J, will be 36, 32, 25, multiplied by 7 {■= 2" — i) to the fame Denominator; and their Sum taken from the Fines to be paid after one Game is won, which are 6 = 2" — 2, leaves for the Rifks of B, -^^ : thofe of C ' '49 ' D, A, being -^, -~^ , -— , refpedively. 3. If 5 has got more than one Game, the Sums for which a Spedlator R may furnilh all the fubfequent Fines^ will be found as follows. Let 158 Ihe Doctrine c/ Chances. Let the Number of Finei which R rifks to pay, when B has got I, 2, 3, 4, &c. Games, be x, y, z, v, ficc. refpedlively ; then . -r -r.)-r —-x; or y = x — 2, — ■ — =zy =x — 2 ; or z-^= X — 2^-1-2. And the lame way v=^x — z^-\-z'^-{-2, &c. ; in an obvious Progreffion. Becaufe when B has got i Game, there is an equal Chance of his winning or lofing the next ; in the former Cafe, R pays the Fine 1 Xp for C, and comes to have the Rifk j; but if C wins, R pays 1 Xp for B, and his Rifk x is the fame as before : and fo of the reft. So that the number of pieces p for which R may engaa;e to furnifli the fubfequent Fines, when B has got 2, 3,4, &c. Games, is had />y the continual Suit ration of 2 a?7d its Powers from 2" — 2. As in a Poule of four, when B has got 2 Games, the Sum of the Rifks is 6 — 2 =1= 4. In a Poule of five, ,v = 2 • — -2 = 14, y^= 1 2, ;?; = 8, "u = o. And from thefe numbers fubtrading the Rifiisof the other G;"me- flers C, D, E, &c. found as above, there will remain the Rilks of B the Gamefter who continues in play. 4. The Expectations of the feveral Gameflers upon the Fines may likewife be determined by an obvious, but more troubiefome, Operation. Under the Capitals, B C D E A, write their fmall Letters thus : L b' c' d' e' of II. b" c' d" e" a" III. b'" c'" d" e'" a"' IV. b' o o 00 Signifying, refpeftively, the Number of Fines which a Gamefter, winning the Pou/e, may expedl to find in it, B having already got fo many Games as the Dots affixed to the Letter : and to thefe Letters prefix their fractional Co- efficients taken from the Tables of Probabilities. Then, by the law cf the Game, there will be formed a Series of Equations determining the Expedations fought. As in the Cafe of 3 Gamefters, write, B C A I. ±1,1 Ic' la' , , ^ ' " l>and the Equations! 2°. -^c'— - x - X ^' -^ 1 -\- o lUxb"=.2 o o \ ^ -^— — (3°.l^'=lx^X^'+i+o Which I°.f^'=^X2+^X^'4-I The Doctrine ^Chances. IJ9 Which being reduced give £' (= 1/5') = — , O (= ^c') = ~, A' {^= ^a')=z-^. From which fubtradling their refpedlive Rifks ; for B, ^ J for C, -11 ; and for ^, -I^ (= 1 + i, his Fines, and the Fine already paid in), remain the Gains A — ^— . -\ — — — . ' 49 > I 49 > -— , multiplied into p. If it is a Poule of four, the Expectations on the Fines will ftand thus: B C D A 149 '49 '49 I. 87 II. -^h" '49 M9 l^c" -^d" —a" ' 49 1 : 9 III. I X ^"' = 30 o o and fetting afide the common Denominator 149, the Equations will be; 1°. 56^'=^ X25.ZK1 i-oyA' 5°. 28c"--^x56 ^4T+o /. e. c"=l?'+z 2°- 3 6f ' -^ 7 >^ 5^- ^J -1-16^ 6°. 'W^f '+2 3°. y-d'^~x{(K^\-\-2^c" JO, ^ a"=d'-\-2 4". 2 5^?'=-ix 3 2 .^TT+iW 8°. ~-/5"= - X 3 H-il-ZfT: Whence will be found 81 - ^ ^'.-= ---,and(-^ b'=)B'='-^ 149 V 149 ;•" 22201 149 2 1075 — <^r IIT5 r>r= 149 22 '34'- jand 2j,__ 37600 Z2Z0I "49 ~- i and -tf= ^4 6i8- ^"^-^;and5"^: J49 22201 i;^?oo ■49 2Z20I \ / .49 ,anaL -^^^ /,, '373-J . "d= i;andZ)"=2iZl6 149 22201 r//_ '473 22201 dli2-i,:.6i '49 22201 And the like Computations maybe made for the fuperior Pouksy. the Compofition of the Equations to be reduced being regular and obvious. P R (X i6o llje Doctrine (t/* Chances. PROBLEM XLVI. Of Hazard. To fi?id at Hazard the Advantage of the Setter tipQ7i all Suppofitioits of Main atid Chance. SoLU T I ON. Let the whole Money played for be confidered as a common Stake, upon which both the Cafter and the Setter have their feveral Expedlations ; then let thofe Expedations be determined in the fol- lowing manner. Firjl, Let it be fuppofed that the Main is vii : then if the Chance of the Cafter be vi or viii, it is plain that the Setter having 6 Chan- ces to win, and 5 to lofe, his Expedlation will be — of the Stake : but there being 10 Chances out of 36 for the Chance to be vi, or VIII, it follows, that the Expectation of the Setter refulting from the Probability of the Chance being vi or viii, will be — multiplied by— ^ or — divided by 36. Secondly, If the Main being vii, the Chance (hould happen to be V or IX, the Expectation of the Setter would be -4^ divided by 36- 'Thirdly, If the Main being vii, the Chance fliould happen to be ]v or X, it follows that the Expectation of the Setter would be 4 divided by 36. Fourthly, If the Main being vii, the Cafter fliould happen to throw II, III, or XI r , then the Setter would neceflarily win, by the Law of the Game; but there being 4 Chances in 36 for. throwing II 1 1 1, or X II, it follows that before the Chance of the Cafter is thrown, the Expectation of the Setter refulting from the Probability of the Cafter's Chance being 11, ii i, or xi i, will be 4 divided by 36. Lajily, If the Main being vi i, the Cafter fliould happen to throw VI I, or XI, the Setter lofe s his Expectation. From the Solution of the foregoing particular Cafes it follows, that the Main being vi i, the Expectation of the Setter will be ex- prefled Ihe Doctrine*?/ Chances. i6i 12. q- -if_ _|_ ± + 1 preffed by the following Quantities, viz. — ^ ! L which may be reduced to -^^ ; now this fradlion being fubtradt- ed from Unity, to which the whole Stake is fuppofed equal, there will remain the Expedation of the Cafter, viz. -^ii- . . . . . . +95 But the Probabilities of winning being always proportional to the Expedlations, on Suppofition of the Stake being fixt, it follows that the Probabilities of winning for the Setter and Cafter are refpcdively proportional to the two numbers 251 and 244, which properly de note the Odds of winning. Now if we fuppofe each Stake to be i, or the whole Stake to be 2, the Gain of the Setter will be expreffed by the fraction — - , it being the difference of the numbers expreffing the Odds, divided by their Sum, which fuppofing each Stake to be a Guinea of 2 1 Shillings will be about 'if- — 2 —f. By the fame Method of Procefs, it will be found that the Main being vi or viii, the Gain of the Setter will be -7777 which is about ed. — 2 r/in a Guinea. It will be alfo found that the Main being v or ix, the Gain of the Setter will be ^^ ■ , which is about 3'^- — 3 '-f'v!\2. Guinea. Corollary i. If each particular Gain made by the Setter, in the Cafe of any Main, be refpedlively multiplied by the number of Chances which there are for that Main to come up, and the Sum of the Produdts be divided by the number of all thofe Chances, the Quotient will exprefs the Gain of the Setter before a Main is thrown : from whence it follows that the Gain of the Setter, if he be refolved to fet upon the firft: Main that may happen to be thrown, is to be eftimated by 7728" ~^ z^i\' > ^^ whole to be divided by 24, which be- 495 ing reduced will be _iZ_ , or about 4-/ — 2 7/ in a Guinea. Corollary 2. The Probability of no Main, is to the Probability of a Main as 109 4" 2 to 109 — 2, or as 1 1 1 to 107. Y CoRo- i62 Th^ Doctrine <?/* Chances. Corollary 3. If it be agreed between the Carter and Setter, that the Main fliall always be vii, and it be farther agreed, that the next Chance hap- pening to be Ames-ace, the Carter fliall lofe but half his Stake, then the Carter's Lofs is only - ^'^^ of his Stake, that is about —/in a Guinea. Corollary 4. The Main being vi or viii, and the Carter has j of his money returned in cafe he throws Ames-ace, what is his Lofs ? And if the Main being v or ix, and he has — of his Money returned in cafe he throws Ames-ace, what is his Lofs ? In anfwer to the firrt, the Gain of the Setter or Lofs of the Carter is -r- . 37 In anfwer to the fecond the Lofs of the Carter would be but 782 -^ 29 Corollary 5. If it be made a rtanding Rule, that whatever the Main may hap- pen to be, if the Carter throws Ames-ace immediately after the Main, or in other words, if the Chance be Ames-ace, the Carter fliall only lofe - of his own Stake, then the Play will be brought fo near an Equality, that it will hardly be dirtinguifhable from it ; the Gain of the Carter being upon the whole but -^— of his own Stake, or -^ of a farthing in a Guinea. The Demonrtration of this is eafily deduced from what we have faid before ^'/z. that the Lofs of the Carter is -~^ ; now let us confider what part of his own Stake fhould be returned him in cafe he throws Ames-ace next after the Main ; Let z be that part, but the Probability of throwing Ames-ace next after the Main is -^ , therefore, the real Value of what is returned him is -~z, and fince the Play is fuppofed to be reduced to an Equality, then what is returned him murt equal his Lofs ; for which reafon, we have the Equation ~ = -^, or;.=:^ which being very near 7!5<? Doctrine <?/"Chances. 163 near - , it follows that - of his own Stake ought to be returned him. Or thus ; if the Carter has returned him — when that happens, he lofes nothing ; but there being but i Chance in 36 for that Cafe to happen ; the real Value of what is returned is but ^"^ ^ -, and in the fame manner if — is returned, the real Value is ,^ . : and i'o, the Difference — ^ — — r= — ^r is the Gain of the 3X36 50x30 t.048 Carter. PROBLEM XLVII. To fold at Hazard the Gain of the Box for any number oj Games divijible by 3. Solution. Let a and b refpedlively reprefent the Chances for winning a Main ■or for lofing it, which is ufually called a Main and no Mai?i ; then, 1°, It is very vifiblethat when the four laft Mains are ^^fi^, other- wife that when a Main has been lort, if the three following Mains are won fucceffively, then the Box muft be paid. 2°, That the lart 7 Mains being baaaaaa^ there is alfo a Box to be paid. 3°, That the laft 10 Mains being baaaaaaaaa, the Box is to be paid, and fo on. Now the Probability of the 4 laft Mains being baaa is and confequently, if the number of Mains thrown from the beginning is reprefented by ;;, the Gain of the Box upon this account will be But to obviate a difficulty which may perhaps arife concerning the foregoing Expreflion which one would naturally think muft be — ^ — , it muft be remembered that the Termination baaa be- longs to 4 Games at leaft, and that therefore the three firft Games are to be excluded from this Cafe, tho' they ftiall be taken notice of afterwards. Y 2 Again 164 T'he. Doctrine ©/'Chances. Again the Probability of the 7 laft Mains terminating thus baaaaaa^ will be — 1 ■ , but this Cafe does not belong to the 6 firft Mains, a-x-oV therefore the Gain of the Box upon this account will be -— f=r" ' and 10 on. And therefore the firft part of the Expedation of the Box is ex- preffed by the Series of which the number of Terms is "~^ . 3 The fecond part of the Expedlation of the Box arifes from all the Mains being won fucceflively without any interruption of a no Main, and this belongs particularly to the three firft Mains, as well as to all thofe which are divilible by 3, and therefore the fecond part of the Expedation of the Box will be expreffed by the Series -=^— 4- -1., -1- -L, 4- -1. , &c. of which the number of Terms is — . 3 Thofe who will think it worth their while to fum up thefe Series, may without much difficulty do it, if they pleafe to confult my Mif- cellanea, wherein fuch forts of Series, and others more compound, are largely treated of. In the mean time, I fhall here give the Refult of what they may fee there demonftrated. U the firft Series be diftinguifhed into two others, the firft pofitive, the other negative, we fliall now have three Series, the Sums of which will be, fuppofing ; - ■=. r, nh r< — »! X a+i ^ ,_^! 1 » 1 ' — 2» 3^ ,, 3 ^3 , »+3 the fum of all which will be reduced to the Expreffion ~ — 14 49 _i- — i — , when a and ^ are in a Ratio of Equality. 49x2" Corollary i. If n be an infinite number, the Gain of the Box will be univer- fally expreffed by ^^^ x ■ _-— ^j - — j but when a and if are in a. Ratio of Equality by -^ . C o r o- The Doctrine ^Chances. 165 Corollary 2. The Gain of the Box being fiich as has been determined for an infinite number of Mains, it follows that, one with another, the Gain of the Box for one lingle Main ought to be eftimated by — 7— ■ X — r-p, , or T~ if ^ anti ^ are equal. Corollary 3. And confequ entlv, it fo llows that in fo many Mains as are ex- preffed by -l±iil±^^iz£_ , or in 14 Mains if a and b are equal, the Expedation of the Box is i, calling i whatever is ftipulated to belong to the Box, which ufually is 1 Half-Guinea. Coroll ar y 4, Now fuppofing that a and 1^ are refpedively as 107 to iii, a Box is payed one with another in about 14.7 Mains. After 1 had folved the foregoing Problem, which is about 12 years ago, 1 fpoke of my Solution to Mr. Henry Stuart Stevens, but with- out communicating to him the manner of it : As he is a Gentleman who, befides other uncommon Qualifications, has a particular Sagacity in reducing intricate Queftions to iimple ones, he brought me, a few days after, his Inveftigation of the Conclufion fet down in my third Corollary ; and as I have had occafion to cite him before, in another Work, fo I here renew with pleafure the Expreffion of the Efteem which I have for his extraordinary Talents : Now his Invef- tigation was as follows. Let a and b refpedively reprefent the number of Chances for a; Main and no Main ; Let alfo i be the Sum which the Box muft re- ceive upon Suppofition of three Mains being won fucceffively ; now the Probability of winning a Main is -^^ , and the Probability of winning three Mains is ~=^pr j a"<^ therefore the Box-keeper might without advantage or difadvantage to himfelf receive from the Cafter at a certainty, the Sum ■ - "' . x i, which would be an Equi- valent for the uncertain fum J , payable after three Mains. Lat i66 Hhe Doctrine o/" Chances. Let it therefore be agreed between them, that the Carter fliall pay but the Sum ■ - "—., X i for his three Mains; now let us fee what con- fideration the Box-keeper gives to the Carter in return of that Sum. 1°, he allows him one Main fure, 2°, he allows him a fecond Main conditionally, which is provided he wins the firft, of which the Probability being -^rf > it follows that the Box allows him only, if one may fay fo, the portion -^^ of a fecond Main, and for the fame reafon the portion JZ^- of a third Main, and therefore the Box allows in all to the Carter i + -^ -}- -:^^ Mains, or ^aajr^oh-^hb ^^^ therefore if for the Sum received -=-^r X i, there be the allowance of """" ;^ ^ Mains, how many are a + I)]' allowed for the Sum i ? and the Term required will be 3«. + 3.^ +£^x7Tr ^j. T^ _ j>H . 3nd therefore in fo ma- ny Mains as are denoted by the foregoing Expreflion, the Box gets the Sum I ; which Expreflion is reduced to 14 if ^z and b are equal. PROBLEM XLVIII. Of Raffling. If any number of Gamejlers A, B, C, D, &'c, play at Raffles, what is the Probability that the Jirji of them having thrown his Chance, and before the other Chalices are throw?i^ wins the Money of the Play ^ Solution. In order to folve this Problem, it is necelTary to have a Table ready compofed of all the Chances which there are in three Raffles, which Table is the following. Ji *The Doctrine o/" Chances. 167 A Table of all the Chances which are in three Rajfki. \^fwr\^^ Chances to Chances to Equality of jruinis* winorlofe. winorlofe. Chance. LTV "1 nx 884735 I LIII X 884726 I 9 LII XI 884681 10 45 LI XII 884534 ss H7 L XIII 884165 202 369 XLIX XIV 883400 57^ 765 XLVIII XV 881954 1336 1446 XLVII XVI 879470 2782 2484 XL VI XVII ^ISS^"^ 5266 3969 XLV XVIII 869632 Q235 586Q XLIV XIX 861 199 15104 8433 XLIII :> or < XX 849706 23537 "493 XLII XXI 834679 35030 15027 XLI XXII 815392 50057 19287 XL XXIII 791506 69344 23886 XXXIX XXIV 762838 93230 28668 XXXVIII XXV 728971 121898 33867 XXXVII XXVI 690100 ^SSl^S 38871 XXXVI XXVII 646929 194636 43171 XXXV XXVIII 599472 237807 47457 XXXIV XXIX 548865 285264 50607 XXXIII XXX 496314 335871 52551 XXXII LXXXI 442368 388422 53946 The Sum of all the numbers expreffing the Equality of Chance being 442368, if that Sum be doubled it will make 884736, which is equal to the Cube of 96. The firft Column contains any number of Points which A may be fuppofed to have thrown in three Raffles. The fecond Column contains the number of Chances which A ha& for winning, if his Points be above xxxi, or the number of Chances he has forlofing, if his Points be either xxxi or below it. The third Column contains the number of Chances which A has for lofing, if his Points be above xxxi, or for winning, if they be either XXXI or below it. The 1 68 The Doctrine <?/" Chances. The fourth Column, which is the principal, and out of which the other two are formed, contains the number of Chances whereby any jiuniber of Points from ix to liv can be produced in three Raffles; and confequently contains the number of Chances which any of the Gamellers 5, C, Z), Sec. may have for coming to an equality of Chance with A. The Conftruftion of the fourth Column depends chiefly on the number of Chances which there are for producing one fingle Raffle, whereof xviii or III have I Chance xvn or IV have 3 Chances XVI or V have 6 Chances XV or v[ have 4 Chances XIV or VI I have 9 Chances XIII or VIII : have 9 Chances XII or IX have 7 Chances XI or X have 9 Chances Which number of Chances being duly combined, will afford all the Chances of three Raffles. But it will be convenient to illuftrate this by one Inftance ; let it therefore be required to find the number of Chances for producing X 1 1 Points in three Raffles. 1°, It may plainly be perceived that thofe Points may be produ- ced by the following fingle Raffles 1 1 1, 1 1 t, vi, or 1 1 1, iv, v, or IV, IV, IV; then confidering the firft Cafe, and knowing from the Table of fingle Raffles, that the Raffles i 1 1, in, vi, have refpec- tively I, 1,4 Chances to come up, it follows from the Dodlrine of Combinations that thofe three numbers ought to be multiplied toge- ther, which in the prefent Cafe makes the produdl to be barely 4, but as the difpofition, in, in, vi, may be varied twice; viz. by III, VI, III, and VI, in, iii, which will make in all three dif- pofitions, it follows that the number 4, which exprefl^es the Chances of one difpofition, ought to be multiplied by 3, which being done, the produdt 1 2 muft be fet apart. 2°, The Difpofition ni, iv, v, has for its Chances the pro- du6t of the numbers i, 3, 6, which makes 18; but this being capable of 6 permutations, the number 18 ought to be multiplied by 6, which being done, the produdl 108 muft likewife be fet apart. 3°, The Difpofition iv, iv, iv has for its Chances the produd of 3, 3, 3, which makes 27 ; but this not being capable of any varia- tion, we barely write 27, which muft be fet apart. 4% The Doctrine 0/ Chances. 169 4% Adding together thofe numbers that were feverally fet apart, the Sum will be found to be 147, which therefore expreffes the num- ber of Chances for producing xii Points in three Raffles: and in the fame manner may all the other numbers belonging to the Table of three Raffles be calculated. This being laid down, let us fuppofe that A has thrown the Points XL in three Raffles, that there are four Gameflers befides himfelf, and that under that circumflance of Ji^ it be required to find the Pro- bability of his beating the other four. Let tn univerfally reprefent the number of Chances which any other Gamefler has of coming to an equality with A^ which number of Chances in this particular Cafe is 23886; Let a univerfally reprefent the number of Chances which A has for beating any one of his Adverfaries, which number of Chances is found in the Table to be 791506; Let/ reprefent the number of all the Chances that there are in three different Raffles, which number is the Cube of 96, by reafon that there are no more than 96 fingle Raffles in three Dice, and therefore /' conflantly flands for the number 884736; Let p univerfally reprefent the number of Gameflers in all, which in this Cafe will be 5 ; then the Probability which A has of beating the other four will be j—i — ; and therefore if each of the mpYp Gameflers flake i, the Expedation of -4 upon the whole Stake/, will be expreffed by --:-; ; and confequently his Gain, or what he might clearly get from his Adverfaries by an equitable compofition with them for the Value of his Chance, will be m/- — I. Now the Logarithm oia -f- ;« = 5.9 1 13665, Log. a =5.8984542, Log- m ■=■ 4.3781 434, Log. /^ 5.9468136 ; and therefore Log. a-\-m P = or Log. a -\- niV = 29.5568375, Log. «? =29 4922710, Log. ;///?—' =28.1653978; from which Logarithms it will be conve- nient to rejedl the leafl index 28, and treat thofe Logarithms as if they were refpedively 1.5568325, 1.4922710, 0.1653978: but the num- bers belonging to the two hrfl are 36,044 and 3 J. 065, whole diffe- rence is 4.979 from the Logarithm of which, viz. 0.697 142 i, ^^ the Log. 0.1653978 be fubtradled, there will remain the Log. °-53 ^7433' of which the correfponding number being 3.402, it fol- lows that the Gain of ^ ought to be eflimated by 2.402. Z Demon- lyo Ihe Doctrine 6/" Chances. Demon strat i on. 1°, When A has thrown his Chance, the Probability of i3's hav- ing a worfe Chance will be j ; wherefore the Probability which A has of beating all his Adverfaries whofe number \% p — i, will be /— 2°, The Probability which B has in particular of coming to an Equality with ^ is y , which being fuppofed, the Probability which A has of beating the. reft of his Adverfaries whofe number it p — 2j is ■ " ■ ; which being again fuppofed, the Probability which A now has of beating B, with whom he muft renew the Play, is — ) wherefore the Probability of the happening of all thefe I p—2 J--2- , —ma things Is r= y X -^—^ X -z=.—^^ — : but becaufe C, or D or E, &c. might as well have come to an equality with A 2iS B him- felf, it follows that the preceding Fradlion ought to be multiplied by * — I, which will make it, that the Probability which A has of beating all his Adverfaries except one, who comes to an equality with him, and then of his beating him afterwards, will be p—\ p—i — ma 3°, The Probability which both B and C have of coming to an equality with A is -^"- ; which being fuppofed, the Probability which A has of beating the reft of his Adverfaries whofe number is p — o^ is " ; which being again fuppofed, the Probability which A now has of beating B and C with whom he muft re- new the Play, (every one of them being now obliged to throw for a new Chance) Is 4 ; wherefore the Probability of the hap- pening of all thefe things will be = -^ ^ ^_ x — ==: —mma -^ — : but the number of the Adverfaries of ./^ being /> — j, and The Doctrine (9/"Chances. 171 and the different Variations which that number can undergo by elec- tions made two and two being — j — x — , as appears from the Dodtrine of Combinations, it follows that the Probability which any two, and no more, of the Adyerfaries of A have of coming to an Equality with him, that ^4 fhall beat all the reft, and that he fhall beat afterwards thofe two that were come to an Equality, is X — mma y^-' • and fo of the reft. From hence it follows that the Probability which ^ has of beit- ing all his Adverfaries, will be expreffed by the following Series, P — ' I ?~' " — 2 , ^— I p—z p — 3 ■ fi— 1 p—2 p — 0, -I *_- /-' the Terms of whofe Numerator are continued till fuch time as their number be =/); now to thofe who underftand how to raife a Bino- mial to a Power given, by means of a Series, it will plainly appear that the foregoing Expreffion is equivalent to this other '^ '" which confequently denotes the Probability required. PROBLEM XLIX. The fame things being given as in the preceding Problem^ to find how many Gamefiers there ought to be in all^ to make the Chance of A, after he has thrown the Point XL, to be the moft advantageous that is pojfible. Solution. It is very eafily perceived that the more Adverfaries A has, the more his Probability of winning will decreafe ; but he has a Com- penfation, which is, that if he beats them all, his Gain will be greater than if he had had fewer Competitors : for which reafon, there being a balance between the Gain that he may make on one fide, and the decreafe of the Probability of winning on the other, there is a certain number of Gamefters, which till it be at- tained, the Gain will be more prevalent than the decreafe of Proba- bility ; but which being exceeded, the decreafe of Probability will fH-evail over the Gain ; fb that what was advantage, till a certain time, may gradually turn to equality, and even to difadvantage. This Problem is therefore propofed in order to determine thofe Circum- ilances. Z 2 Let 172 The Doctrine ^Chances. Let Log./ — Log. a be made=^, letalfo Log./ — Log. a-\-m be made =J, which being done, then the number of Gamefters re- quifite to make the Advantage the greateft poflible will be exprefled bv the fradlion , "^J^ "^ fo that fuppofing as in the pre- ceding Problem that ^2 = 791506, ;«= 23886, and confequently rt -|- W3=8i5392, as alfo /= 884736, and Log. / = 5.9468136 Log.^ = 5.8984542,Log.;;z=4.378i434Log.fl + w=:5.9ii366^, then ^ will be = 0.0483594, and / will be= o. 3 5447 1 . Theref. Log. g ■ — ■ Log./':= o. 1 3490 14, and Log. a -\- m — Log. ^=0.01 ^ 9 1 23 and therefore the number of Gamefters will be --/igTii^'" == ^ °4 nearly, which fliews that the number required will be about ten or eleven. As the Demonftration of this laft Operation depends upon prin- ciples that are a little too remote from the Dodlrine of Chances, I have thought fit to omit it in this place -, however if the Reader will be pleafed to confult my Mifcellanea Analytica, therein he will find it, pag. 223 and 224. It is proper to obferve, that the method of Solution of this laft Problem, as well as of the preceding, may be applied to an infinite variety of other Problems , which may happen to be fo much eafier than thefe, as they may not require Tables of Chances ready calcu- lated. PROBLEM L. Of Wh 1 S K. If four Gameflers play at TVhiJk, to find the Odds that any two of the Partners, that are pitched upon, have not the four Hojiours, Solution. F/r/?, Suppofe thofe two Partners to have the Deal, and the laft Card which is turned up to be an Honour, From the Suppofition of thefe two Cafes, we are only to find what Probability the Dealers have of taking three fet Cards in twenty- five, out of a Stock containing fifty-one. To refolve this the fhorteft way, recourfe muft be had to the Theorem given in the Remark be- longing to our xx^"" Problem, in which making the Quantities ;/, iT^e Doctrine <?/^ Chances. 17;^ c,d,p, a, refpedlively equal to the numbers 51, 25, 26, 3, 3, the Probability required will be found to be -^-^iii^ii^ or ■ ^" . Secondly, If the Card which is turned up be not an Honour, then we are to find what Probability the Dealers have of taking four given Cards in twenty-five out of a Stock containing fifty- one ; which by the aforefaid Theorem will be found to be """'^''"''" ' or 5 I X50X49X4S 4998 But the Probability of taking the four Honours being to be efti- mated before the lafl Card is turned up ; and there being fixteen Chances in fifty-two, or four in thirteen for an Honour to turn up, and nine in thirteen againft it, it follows that the Probability of the firft Cafe ought to be multiplied by 4 ; that the fradion exprefling the Probability of the fecond ought to be multiplied by 9 ; and that the Sum of thofe Produdts ought to be divided by 13, which being done, the Quotient -^-q- or -^ nearly, will exprefs the Probability required. And by the fame Method of proceeding it will be found, that the Probability which the two Eldeft have of taking four Honours is —^^l^ , that the Probability which the Dealers have of taking three Honours is -^^ , and that the Probability which the Eldeft have of taking three Honours is ^, ^ . Moreover, that the Probability that there are no Honours on either fide will be .,^^ . Hence it may be concluded, 1°, that it is 27 to 2 nearly that the Dealers have not the four Honours. That it is 23 to I nearly that the Eldeft have not the four Honours. That it is 8 to I nearly that neither one fide nor the other have the four Honours. That is 1 3 to 7 nearly that the two Dealers do not reckon Honours. That it is 20 to 7 nearly that the two Eldeft do not reckon Honours. 'And that it is 25 to 16 nearly that either one fide or the other do reckon Honours, or that the Honours are not equally divided. Corollary i. From what we have faid, it will not be difficult tofolve this Cafe at Whifk ; viz. which fide has the bcft, of thole who have viii of tiie Game, or of thofe who at the fame time have ix ? In J74 ^^ Doctrine <?/ Chances. In order to which it will be neceflary to premife the following Principle. 1°, That there is but i Chance in 8 192 to get vi i by Triks. 2°, That there are 13 Chances in 8192 to get vi. 3°, That there are 78 Chances in 8 192 to get v. 4°, That ihere are 286 Chances in 8192 to get iv. 5°, That there are 715 Chances in 8192 to get iii. 6°, That there are 1287 Chances in 8192 to get 11. 7°, That there are 17 16 Chances in 8192 to get i. All this will appear evident to thofe who can raife the Binomial ^z -4- ^ to its thirteenth power. But it mufl: carefully be obferved that the foregoing Chances ex- prefs the Probability of getting fo many Points by Triks, and nei- ther more nor lefs. For if it was required, for Inftance, to aflign the Probability of getting one or more by Triks, it is plain that ihe Numerator of the Fradtion expreffing that Probability would be the Sum of all the Chances which have been written, viz. 4096, and confequently that this Probability would be -—^ or — . -' 8192 2 2°, That the Probability of getting two or more by Triks would be or 8192 ' jrgj 3°, That the Probability of getting three or more by Triks would 8192 4°, That the Probability of getting iv or more by Triks would 5°, That the Probability of getting v or more by Triks would be 9z 8192 6°, That the Probability of getting vi or more would be 14. 8192 ■ 7°, That the Probability of getting v 1 1 would be -g— • . This being laid down, I proceed thus. 1°, If thofe that have v 1 1 1 of the Game are Dealers, their Proba- bility of getting 1 1 by Honours is -j^ : for the Dealers will get 1 1 by Honours if they have either 3 of the 4 Honours, or all the 4 Honours, but the Probability of taking three Honours is -^^ , and the Probability they have of taking the four Honours is -—^ , and the Sum of this is -^j^ . The The Doctrine i?/" Chances. 175 The Probability which they have of getting them by Triks is z?8o ii!.o or 8:92 .< > 96 And therefore adding thefe two Probabilities together, the Sum Will be ,,,3^.6 • Now fubtradling from this, the Probability of both circum- ilances happening together, '•oiz. - ^g^;'" - the remainder will be -3-2-/-i- ; and this exprefles their Expedlation upon the common Stake which we fuppofe to be r=: i . But they have a farther Expedlation, which is that of getting one fin2:le Game by Triks, which is ^-^^ or ■ ''"'^ -, and their Proba- bility of not getting by Honours is ^^— (= i ^7^ ) ; and there- fore their Probability of getting one fingle Game by Triks indepen- dently from Honours is - ^,'^,q°V ; but then if this happen they will be but equal with their Adverfaries, and therefore this Chance entitles them to no more than half of the common Stake ; therefore taking the half of the foregoing fradion, it will be g^^^^^j and therefore the whole Expedation of the Dealers is ^^^2^2^±^^— liiiMi. r bb23936 0823936 whence there remains for thofe who have ix of the Game ,^"^^'"'^' ; 68230^6 » which will make that the Odds for the vi 1 1 againft the ix will be 4141345 to 2b82i|9i, which is about 3 to 2, or fomething more, 17/2;, r; to II. 2*, But if thofe who have v 11 1 of the Game are Eldeft, then their Probability of having three of the four Honours is —^ , and their Probability of having the four Honours is -73"— , and therefore their Probability of getting their two Games by Honours is , ,: , -^-r^Q __ ■ '^' ^' ■ . 7 he Probability of getting them by Triks is as before -^-^^ , now adding thefe two Probabilities tog'ether, the Sura will be •^i~r^ , from which fubtrading, the Probability of both circumftan- O ..L . - ...Ml • •■.-X~'<x9, ces happening together, viz. ' '"'"'^^ , there vi'ill remain ^ ^ DO '■■ .1930 c.i?.39,; * and this expreffes the Expedation arifing from the Profped of their winning at once either by Honours or by Triks. But 176 The Doctrine (j/' Chances. But their Expectation arifing from the Profped of getting one fin- gle Game, and then being upon an equal foot with their Ad veifaries, found the fame way as it was in the Suppofition of their being Dealers, is "ll ' ^"^^ - . For the Probability of tlic Elded taking 4 Honours is -y^ , and of their taking 3 Honours, -^ ; whofe Sum taken from Unity, leaves -|^^ , for the Probability of their not getting by Honours ; and this multiplied by ~^ the Probability of their get- ting one Game by Triks, gives -^-7^^; the half of which is ' blilllb ' '^"'^ therefore their Expedtation upon the whole is '^hflj^lp-^-- = "Slifr ' ^"^ confequently there remains for the IX, -^——^t and therefore the Odds of the viii againft the ix are now 3769795 to 3054141, which is nearly as g^ to yy. From whence it follows that without confidering whether the VIII are Dealers or Eldeft, there is one time with another the Odds of fomewhat lefs than 7 to 5; and very nearly that of 25 to 18. Corollary 2. It is a Queftion likewife belonging to this Game, what the Proba- bility is that a Player has a given number of Trumps dealt him : par- ticularly, it has been often taken as an equal Wager that the Dealer has at leaft 4 Trumps. Now altho' the Solution of all fuch Queflions is included in our xx'*" Problem ; yet as this Game is much In ufe, I have, for the Reader's eafe, computed the following Tables ; fhewing, for the Dealer as well as the other Gamefters, what the Probability is of taking precifely any afligned number of Trumps in one deal. And thence by a continual addition of the numbers, or of fuch part of them as is neceflary, it is ealily found what the Probability is of taking at leaji that number. Chances The Doctrine <?/" Chances. 77 Chances of the Dealer to have befides the Card turned up. Trumps Chances of any other Game- fler to have precifely. 3910797436 201 12672528 41959196136 46621329040 I. II. III. 8122425444 46929569232 1 10619698904 139863987120 30454255260 121817^2104 3014663652 IV. V. VI. 104897990340 48726808416 14211985788 455999544 Tab. I. 40714245 VII. VIII. 25*^3997416 284999715 Tab. II. 2010580 IX. 1.8095220 48906 468 I X, XI. XII. 603174 8892 39 Sum — 158753389900 is the 476260169700 .. Sum, is] commom Denominator ; be- the common Denominator ; ing the Combinations of j 2 being the Combinations of Cards in 51. 13 in iji. By the help of thefe Tables feveral ufeful Queftions may be re- folved ; as 1°. If it is afked, what is the Probability that the Dealer has precifely III Trumps, befides the Trump Card ? The Anfwer, by T^ab. I. is 4662 J and the Probability of his having fome other But if the Qucftion had been, What \\2\\ 15675 '5875 number of Trumps is is the Probability that fome other Gamefter, the eldeft hand for inflance, has precifely IV Trumps? The anfwer, by Tab. II. is 104SQS 476263 2°. To find the Chance of the Dealer's not having fewer than IV Trumps: add his Chances to take o, I, II, which a^e 39108, 201127, 4i9i;92; and their Sum 659827 taken from the Deno- minator 1587534, and the Remainder made its Numerator, the Probability of the Dealer having IV or more Trumps will be - '"^ °l- a little above — r . The Wager therefore that the Dealer 32Q 12 has not IV Trumps is fo far from equal, that whoever lays it throws away above ^ of his Stake. A a Bi;t 178 n^e Doctrine o/" Chances. But if the Wager is that the Dealer has not V Trumps, there 466213 (the Chances of his having III. befides the Trump Card) is to be added to the Chances for o, I, II ; which will make the Chance of him who lays this Wager to be nearly -iil-; and that of his 45 S Adverfary . 455 And hence, if Wagers are laid that the Dealer has not IV Trumps, and has not V Trumps, alternately; the advantage of him who lays in this manner will be nearly 1 1 — per Cent, of his Stakes. 3°. To find the Odds of laying that the eldeft hand has at leaft III, and at leaft IV Trumps, alternately, the Numerator of the one Expeftation is (by T^^. II.) 31 501 119, and of the other 175 14720, to the Denominator 47626017 ; whence the advantage of the Bet will be — ^ , or 3 per Cent, nearly. Again, if it is laid that the Trumps in the Dealer's hand fhall be either I, II, 111 or VI; the difadvantage of this Bet will be only 15 A 4"/, or about -7 , -per Cent. In like manner, the Odds of any propofed Bet of this kind may be computed : And from the Numbers in the Tables, and their Combi- nations, different Bets may be found which (hall approach to the Ratio of Equality; or if they differ from it, other Bets may be affigned, which, repeated a certain Number of Times, fhall ballance that difference. 4", And if the Bet includes any other Condition befides the num- ber of Trumps, fuch as the Quality of one or more of them ; then proper R.egard is to be had to that reftridlion. Let the Wager be that the Eldefl has IV Trumps dealt him ; and that two of them fhall be the Ace and King. The Probability of his having IV Trumps precifely is, by Tab II. ~^ : and the dif- ferent fours in 1 2 Cards are -7- X ~- X -j- X |- . But becaufe 2 out of the 12 Trumps are fpecified, all the Combinations of 4 ia 1 2 that are favourable to the Wager are reduced to the different two's that are found in the remaining 10 Cards, which are — X7. And this number is to the former as i to 11: the Probability therefore is reduced by this reftridion to -^ , of what elfe it had been : that is, it is reduced from near j to about -^ . ^Vo^^ ; *The Doctrine (^Chances. 179' Note; thefe Tables and others of a like kind, which different Games may require, are befl computed and examined by beginning with the loweft number, and obferving the Law by which the others are formed fuccelfively. As in Tab. I, putting A^=i; and the Let- ters B, C, Z), &c. ftanding for the other Terms regularly afcending ; we fhall have B=-^x-^ycA C=-^x-^x5, Z) = -^x — X C, &c. till we arrive at the Term iV=: -^ x -^ X M 3 1212 And if the correfponding Terms in Tab. II. are marked by the fame Letters dotted, then is yf' = ^ x ^, S' = ^ X fi, C=-^xC, £)'= — X A &c. up to N'=.^X N. PROBLEM LI. OfPiaUET. To find at Piquet the Probability which the Dealer has for taking one Ace or more in three Cards^ he having none in his Hand^ Solution. From the number of all the Cards which are thirty-two, fub- tra<Sing twelve which are in the Dealer's Hands, there remain twenty, among which are the four Aces. From which it follows that the number of all the Charlces for taking any three Cards in the bottom, is the number of Combinations which twenty Cards may afford being taken three and three ; which by the Rule given in our xv Problem is / ■ or 1 140. The number of all the Chances being thus obtained, find the number of Chances for taking one Ace precifely with two other Cards ; find next the number of Chances for taking two Aces pre- cifely with any other Card; laflly, find the number of Chances for taking three Aces ; then thefe Chances being added together, and their Sum divided by the whole number of Chances, the Quotient will exprefs the Probability required. A a 2 But i8o The Doctrine ©/"Chances. But the number of Chances for taking one Ace are 4, and the number of Chances for taking any two other Cards, are — . — , and therefore the number of Chances for taking one Ace and two other Cards are — x ■ ', ' 'J - = 480, as appears from what we have faid in the Doftrine of Combinations. If there remains any difficulty in knowing why the number of Chances for joining any two other Cards with the Ace already taken is — — ^ , it will be eafily refolved if we confider that there bein^ in the whole Pack but 4 Aces and 28 other Cards, out of which other Cards, the Dealer has 1 2 in his Hands, there remain only 16, out of which he has a Choice, and therefore the num- ber of Chances for taking two other Cards is what we have de- termined. In like manner it will appear that the number of Chances for taking two Aces precifely are -\~- or 6, and that the number of Chances for taking any other Card are — or 16; from whence it follows that the number of Chances for taking two Aces with an- other Card are 6 X 16 or 96. Laflly, it appears that the number of Chances for taking three Aces is equal to ■-' ; ' \ ' - - = 4. 1.2.3 Wherefore the Probability required will be found to be ^?o4-gf+2.oi._^or-^, which fradion being fubtrafted from 1140 I 140 57 Unity, the remainder will be — . From whence it may be concluded that it is 29 to 28 that the Dealer takes one Ace or more in three Cards, he having none in his Hand. The preceding Solution may be contraded by inquiring at firft what the Probability is of not taking any Ace in three Cards, which may be done thus. The number of Cards in which the four Aces are contained be- ing twenty, and confequently the number of Cards out of which the four Aces are excluded being fixteen, it follows that the num- ber of Chances which there are for the taking of three Cards, among which no Ace fliall be found, is the number of Combinations which fixteen Cards may afford being taken three and three, which num^. bee Tlje Doctrine (?/"Chances. i8i ber of Chances by our 1 8''' Problem will be found to be ', ' ^l \ '^ or 560. But the number of all the Chances for taking any three Cards In twenty has been found to be 11403 from whence it follows that the Probability of not taking any Ace in three Cards, is -j-|-^ or — ::- , and therefore the Probability of the contrary, that is of taking one Ace or more in three Cards is — as we had found it before. 57 PROBLEM LIT. To find at Piquet the Probability which the Eldeji has of taking an Ace i?i five Carets^ he having no Ace in his Hand. Solution. Firjl, Find the number of Chances for taking one Ace and four other Cards, which will be 7280, Secondly, The number of Chances for taking two Aces and three other Cards, which will be found to be 3360. Thirdly, The number of Chances for taking three Aces and two other Cards, which will be found to be 480. Fourthly, The number of Chances for taking four Aces and any other Card, which will be found to be 1 6. Lajlly, The number of Chances for taking any five Cards ia twenty, which will be found to be 15504. Let the Sum of all the particular Chances, viz. 7280 -]- 3360 -|- 480-}- 16, be divided by the Sum of all the Chances, viz. hy 15504,, and the Quotient will be — or —^ — which being fubtraded ^ 15504 323 & from Unity, the remainder will be -2L- • and therefore the Odds of the Eldeft hand taking an Ace or more in five Cards areas 232 to 91, or 5 to 2 nearly. But if the Probability of not taking an Ace in five Cards be in- quired into, the work will be confiderabiy fliortened; for this Pro- bability will be found to be exprelTed by '^'^ '^'"! ' "' or i82 The Doctrine o/" Chances. 4368 to be divided by the whole number of Chances, "v'lz. by 15504, or 9 1 by 3 23 ; which makes the Probability of taking one or more Aces -^^ as before. PROBLEM LIII. *To find at Piquet the Probability which the Eldeji has of taki?jg both an Ace and a Ki?ig in fve Cards^ he havin? no7ie in his Hand. o So LUT I ON. Let the following Chances be found ; viz. 1°, For one Ace, one King, and three other Cards. 2*, For one Ace, two Kings, and two other Cards. 3°, For one Ace, three Kings, and any other Card. 4°, For one Ace, and four Kings. 5°, For two. Aces, one King, and two other Cards. 6°, For two Aces, two Kings, and any other Card. 7°, For two Aces, and three Kings. 8°, For three Aces, one King, and any other Card. 9°, For three Aces, and two Kings. 10°, For four Aces, and one King. Among thefe Cafes, there being four pairs that are alike, •Jiiz. the fecond and fifth, the third and eighth, the fourth and tenth, the feventh and ninth ; it follows that there are only fix Cafes to be cal- culated, whereof the firft and fixth are to be taken fingly, but the fecond, third, fourth and feventh to be doubled j now the Opera- ration is as follows. The/r/? Cafe has ^ X7 X ',' ; ^ ; '° or 3520 Chances. The Jecond -x-^— ^X " " " '" or 15^45 the double of which is 3; 68. The third -y. ■* ' ' x-^ or 102, the double of which is I I . 2 . 3 1 ' 384 Chances, The *The Doctrine ^Chances. 183 The fcurtb - X ^' ' ^ \^ ' ' or 4, the double of which is 8 Chances. The J?xth ^ ' I X ^ " I X -^ or 43 2 Chances. The feventh ^ ' \ X ^ ' ^ ' ^ X or 24, the double of which is 48 Chances. Now the Sum of all thofe Chances being 7560, and the whole number of Chances for taking any five Cards out of 20 being 7 . 7 '. '3 ■ '4 ■ V °'' ^•^5°4' ^'^ follows that the Probability re- quired will be —^ or -^, and therefore the Probability of the 1 5)04 I 4u •' contrary will be -7^, from whence it follows that the Odds a?ainft ^ (/40 O the Eldefl: hand taking an Ace and a King are 331 to 315, or 2 i to 20 nearly. PROBLEM LIV. To find at Piquet the Probability of having twelve Cards dealt tOy without King^ ^een or Knave^ which Cafe is commonly called Cartes Blanches. Solution. Altho' this may be derived from what has been faid in the xx*'* Problem, yet I fliall here prefcribe a Method which will be fome- what more eafy, and which may be followed in many other In- ftances. Let us therefore imagine that the twelve Cards dealt to are taken up one after another, and let us confider, 1°, the ProbabiHty of the firft's being a Blank ; now there being 20 Blanks in the whole Pack, and 32 Cards in all, it is plain that the Probability of it is -^ . 2°, Let us confider the Probability of the fecond's being a Blank, which by reafon the firll: Card is accounted for, and becaufe, there remain now but 19 Blanks and 31 Cards in all, will be found to be -^; and in like manner the Probability of the third Card's being a Blank will be -— , and fo on ; and therefore the Proba- 184 The Doctrine -j/" Chances. Probability of the whole will be expreffed by the Fradion .0 ■ .0 . .3 ■ .7 ■ .^ ■ K ■ .. ■ .3 ■ .z . ,. .^ ■ . thenumberofMul. tiplicators in both Numerator and Denominator being equal to twelve. Now that Fra<flion being fliortened will be reduced to ■ '11^^^ or — ^ — nearly, and therefore the Odds aerainft Carter Blanches are 179 1 to I nearly. P R O B L E M LV. 'To Jind how many different Sets, effentially different from one another^ one may have at Piquet before taking in* Solution. Let the Suits be difpofed in order, and let the various difpofitions of the Cards be written underneath, together with the number of Chances that each difpofition will afford, and the Sum of all thofe Chances will be the thing required. Let alfo the Letters D, H, S, C refpeilively reprefent Diamonds, Hearts, Spades, and Clubs. D, l^Q Doctrine ^Chances. i8s A ^. 5, C Chances. I 0, 0, 4. 8 — . . . 70 2 0, 0, 5. 7 — 448 3 0, 0, 6. 6 = 748 4 0, I, 3» 8 — 448 5 0, I. 4. 7 — 4480 6 0, I, 5» 6 — 12544 7 0, 2, 2, 8 — 784 8 0, 2, 3. 7 — 812544 9 0, 2, 4. 6 — 54880 lo 0, 2, 5. 5 = 87808 II 0, 3. 3. 6 — 87808 12 0, 3. 4> 5 — 219520 13 0, 4. 4> 4 — 343000 14 I, I, 2, 8—: 1792 15 I, I, 3. 7 — 28672 16 I, I, 4. 6 — 125440 17 I, I, 5. 5 — 200704 18 I, 2, 2, 7 — 50176 19 I, 2, 3» 6 — 351232 20 I, 2, 4, 5 - 878080 21 I, 3. 3. 5 — 1404928 22 I, 3. 4. 4 — 2195200 23 2, 2, 2, 6 — 614656 24 2, 2, 3» 5 — 2458624 25 2, 2, 4. 4 — 3851600 26 2, 3. 3. 4 — 6146560 27 3' 3> 3. 3 — 9834496 bum 28,967,278 Which Sum would feem incredibly great, if Calculation did not prove it to be fo. But it will not be inconvenient to {hew by one Example how the numbers exprefling the Chances have been found, for which we mud have recourfe to our xx''' and xxi'*" Problems, and there ex- amine the Method of Solution, the fame being to be obferved in this place. Let it therefore be required to affign the 19''' Cafe, which is for taking i Diamond, 2 Hearts, 3 Spades and 6 Clubs. Then it will eafily be feen that the variations for taking i Diamond are 8, that the variations for taking 2 Hearts are -^-^ — = 28, and that B b ' ' ' the 1 8^ 7^<? Doctrine t/ Chances. the variations for taking 3 Spades are ^ ' ^ ' "^ z=: 56, and that the variations for taking 6 Clubs are — ' I ' ' ^ ' ^ ' I := 28. And 1 .2.3.4.5.0 therefore that the number of Chances for the i g^^ Cafe is the pro- dudl of the feveral numbers 8, 28, 56, 28, which will be found 35'232- There is one thing worth obferving, which is, that when the number of Cards of any one Suit being to be combined together, ex- ceed one half the number of Cards of that Suit, then it will be fuf- ficient to combine only the difference between that number and the whole number of Cards in the Suit, which will make the operation fliorter ; thus being to combine the 8 Clubs by fix and fix, I take the difference between eight and fix, which being 2, I combine the Cards only two and two, it being evident that as often as I take 6 Cards of one Suit, I. leave 2 behind of the fame Suit, and that there- fore I cannot take them oftner fix and fix, than I can take them two and two. It may perhaps feem ftrange that the number of Sets which we have determined, notwithftanding its largenefs, yet fliould not come up to the number of different Combinations whereby twelve Cards might be taken out of thirty-two, that number being 225792840 ; but it ought to be confidered, that in that number feveral fets of the fame import, but differing in Suit might be taken, v^hich would not introduce an eflential difference among the Sets. Remark. Tt may eafily be perceived from the Solution of the preceding Problem, that the number of variations which there are in twelve Cards make it next to impoffible to calculate fome of the Probabi- lities relating to Piquet, fuch as that which refults from the priority of Hand, or the Probabilities of a Pic, Repic or Lurch ; however notwithftanding that difficulty, one may from obfervations often re- peated, nearly eftimate what thofe Probabilities are in themfelves, as will be proved in its place when we come to treat of the reafon- able conjedures which may be deduced from Experiments ; for which reafon I fiiall fet down fome Obfervations of a Gentleman who has a very great degree of Skill and Experience in that Game, after which I fhall make an application of them. Hypotheses. 1 °, That 'tis 5 to 4 that the Eldeft hand wins a Game. 2V The Doctrine (?/"Chances. 187 2*, That is 2 to i, that the Eldeft wins rather without lurch- ing than by lurching. 3°, That it is 4 to i, that the Youngeft Hand wins rather with- out lurching than by lurching. But it muft carefully be obferved that thefe Odds are rertrained to the beginning of a Game. From whence, to avoid Fradions, we may fuppofe that the Eldefl: has 75 Chances to win one Game, and the Youngeft 60. That out of thefe "j^ Chances of the Eldeft, he has 50 to win without Lurch, and 25 with a Lurch. That of the 60 Chances of the Youngeft, he has 48 to win with- out a Lurch, and 1 2 with a Lurch. This being laid down, I fliall proceed to determine the Probabi- lities of winning the Set, under all the circumftances in which A and B may find themfelves. 1°, When A and B begin, he who gets the Hand has the Odds of 6478643 to 3362857 or 23 to 20 nearly that he wins the Set. 2°, If yf has I Game and Bnone. Before they cut for the Hand, the Odds in favour of A arc 682459 to 309067 or 38 to 23 nearly. \i A has the Hand, the Odds are 4627 to 1448, or 16 to 5 nearly. If jB has the Hand^ the Odds in favour of .<^ are 51105810 309067, or 38 to 23 nearly. 3°, If A has I Game, and B i Game. He who gets the Hand has the Odds of 1003 9 to 8186 or 27 to 22 nearly. 4°, \iA has 2 Games and B none. Before they cut for the Hands the Odds are 59477 to 13423, or 3 1 to 7 nearly. If A has the Hand, the Odds are 51 17 to 958, or 16 to 3 nearly. If B has the Hand, the Odds in favour of .^ are 1 151 to 307, or 25 to 7 nearly. 5°, If A has 2 Games and J5 i. Before they cut for the Hand, the Odds are 92 to 43, or 15 to 7 nearly. If u4 has the Hand, the Odds are 11 to 4. * If * In this Cafe 5 has 1 2 Chances for I, and 48 for — , but the number of all the B b 2 Chances i88 Z5j Doctrine o/'Chances. IfB has the Hand, the Odds in favour of A are 17 to 10. 6", If ^ has 2 Games and B 2 Games, he who gets the Hand has 5 to 4 in his favour. I hope the Reader will eafily excufe my not giving the Demon- ftration of the foregoing Calculation, it being fo ealily deduced from the Rules given before, that this would Teem entirely fuper- fluous. PROBLEM LVI. Of Saving Clauses. A /jas 2 Chances to beat B, a?id B has i Chance to beat A ; but there is o?2e Cha7tce which ifititles them both to withdraw their own Stake^ which wefuppofe equal to f ; to find the Gain of A. Solution. This Queftion tho' eafy in itfelf, yet is brought in to caution Be- ginners againft a Miftake which they might commit by imagining that the Cafe, which intitles each Man to recover his own Stake, needs not be regarded, and that it is the fame thing as if it did not exift : This I mention fo much more readily, that fome people who have pretended great fkill in thefe Speculations of Chance have them- felves fallen into that error. Now there being 4 Chances in all, whereof A has 2 to gain f, 'tis evident that the Expedation of that Gain is worth -/; but A having i Chance in 4 to lofe f, the Riik of that is a Lofs which muft be eftimated by —f, and therefore the abfolute Gain of y^ is — / ^-f,ov—f. But fuppofing the faving Claufe not confidered, A would have 2 Chances in 3 to win /J and I Chance in 3 to lofe fy and therefore the Expedation of his Gain Chances between A and B are i ^5, therefore B has ' "^ -^— =z -i-, •'■' '35 — l}5 15. Odds 1 1 to 4. KB has the Hand, then he has 25 for i, 50 for JL=: JilH — _5£, » "'^ . '35 = -^ , Odds 17 to 10. But before they cut for the Hand £ has -±- 4- -H- J- - = -^, Odds 92 to 43. would Hoe Doctrine (j/ Chances. i8g would be worth —f, and the Rifk of his Lofs would be eftimated by ^/; which would make his Gain to be 4"/ 7/=—/ 3 ^ .'- 3 > From whence it may evidently be feen that the condition of drawing Stakes is to be confidcred ; and indeed in this laft Cafe, there are the Odds of 2 to I that A beats B, whereas in the former it cannot be faid but very improperly that A has 2 to 1 the beft of the Game ; for if yf undertakes without any limitation to beat 5, then he muft lofe if the faving Claufe happens, and therefore he has but an equa- lity of Chance to beat or not to beat ; however it may be faid with fonie propriety of Expreffion, that it is 2 to i that A rather beats B than that A beats him. But to make the Queftion more general, let A and B each depo- fite the Sum f; let a reprefent the Chances which A has to beat S, and b the Chances which B has to beat A ; let there be alfo a certain number m of Chances which may be called common, by the hap- pening of which A llvall be entitled to take up fuch part of the com- mon Stake 2/ as may be denominated by the fradion — , and B fliall be entitled to take the remainder of it. Then 1°, it appears that the number of all the Chances being « -{- ^ -f- m^ whereof there are the number a which intitle A to gainy> thence his Gain upon that fcore is „,",j.,„ y^f- 2°, It appears that the number of Chances whereby A may lofe, being b, his Lofs upon that account is -^ . ^ . ,^, x/ 3°, It appears that if the Chances ;;; fliould happen, then y^ would take up the part — of the common Stake if, and thereby gain — / — f or ■ '^~'" x/ But the Probability of the happening of this is — rr-; — ; and therefore his Gain arifing from the Proba- bility of this circumftance is ^ Jj - — x -~-^ %f. From all which it appears that his abfolute Gain is Now fuppofe there had been no common Chances, the Gain o£A would have been "~'' v f. Let it therefore be farther required to aflign what the proportion of fi to r ought to be, to make the Gain of A to be the fame in both Caies. 19^ ^^ Doctrine «>/ Chances. This will be eafily done by the Equation -^^ + T^^S" r= -~ ; wherein multiplying all the Terms by a -\- i> -\- m we fliall have the new Equation a — b -\ ^—^ = — —^^ 2bp — br ■= ra — br, or 2pa Ar 2^/> = 2ra, and therefore pa -\- bp^=.ra, and — r= " - . From which we may conclude, that if the two parts of the common Stake 2/ which A and B are refpedively to take up, upon the happening of the Chances w, are refpedively in the proportion of a to b, then the common Chances oive no advantage to A above what he would have had if they had not exifted. PROBLEM LVII. Odds of Chance and Odds of Money compared. A and B playing together depofit f^- apiece \ K has 2 Chances to win f, and B i Chance to imn f, whereupon A tells B that he will play with him upon aji equality of Chance^ if he B will fet him 2( to if, to which B ajfents : to find whether A has any advantage or dif- advantage by that Bargain. Solution. In the firft circumftance, A having 2 Chances to win / and i Chance to lofe/ his Gain, as may be deduced from the Introdudion, 7f-f I n IS -^ — = - /. In the fecond circumflance, A having i Chance to win 2/ and i Chance to lofe / his Gain is — ^ 3= ^J^ and therefore he gets -/by that Bargain. But if B, after the Bargain propofed, fhould anfwer, let us play upon an equality of Chance, and you fhall flake but \f, and I fhall flake/; and fo I fhall have fet 2 to i, and that A fhould afTent : then he has i Chance to win / and i Chance to lofe -^/, and therefore. his T^e Doctrine ^Chances. igi his Gain is ~ — -^ — r= -^y] and therefore he is worfe by -^ /* than he was in the firft circumftance. But if A, after this propofal of 5, anfvsrers j let us preferve the quan- tity of the whole Stake zf, but do you ftake ^f, and I rtiall ftake -f^ whereby the proportion of 2 to i will remain, and that fi aflents; then A has i Chance to win —/and i Chance to lofe —f^ which makes his Gain to be -^ — ; — ^ = t/ — ~f ^^ ~f which is the fame as in the firft circumftance. And univerfally, A having a Chances to win /, and B having b Chances to win / if they fhould agree afterwards to play upon an equality of Chance, and fet to each other the refpedlive Stakes. ~i^f ^"^ "TTT/^ ^^^" ^^^ G^'" o^ ^ would thereby receive no al- teration, it being in both Cafes "~'', L PROBLEM LVIir. Of the Duration of Play; Two Gameflers A and B whofe proportio?i of fiill is as a to b, each havi?ig a certain number of Pieces^ play to- gether on condition that as often as A wins a Game^ ^fl?all give him one Piece; and that as often as B witis a Game^ A pail give hitn one Piece ; and that they ceafe not to play till fuch time as either one or the other has got all the Pieces of his Adverfary : now let us fuppofe two Spe&ators R and S concerning themf elves about the ending of the Play, the firft of the?n laymiv that the Play will be ended in a certain number of Games which he aff}g7n, the other layi7ig to the con- trary. To find the Probability that S has ofwin?iin<r his wager. SOLU- 192 'the Doctrine o/" Chances. SOLUT I ON. This Problem having ibme difficulty, and it having given me occafion to inquire into the nature of fome Series naturally refult- ing from its Solution, whereby I have made fome improvements in the Method of fumming up Series, 1 think it neceflary to begin with the fimpleft Cafes of this Problem, in order to bring the Reader by degrees to a general Solution of it. Case I. Let 2 be the number of Pieces, which each Gamefter has ; let alfo 2 be the number of Games about which the Wager is laid : now becaufe 2 is the number of Games contended for, \tx. a-\-b be raifed to its Square, -oiz. aa -|- 2ab -\- bb ; then it is plain that the Term zab favours 5, and that the other two are againft him ; and confe- quently that the Probability he has of winning is -==fr . Corollary \{ a and b are equal, neither R or S have any Advantage or Dif- advantage ; but if a and b are unequal, R has the Advantage. Case II. Let 2 be the number of Pieces of each Gamefter, as before, but let 3 be the number of Games about which the Wager is laid : then a -\- b being raifed to its Cube, viz. a"^ -\- ^aab -]- ■i^abb -\- b^^ it will be feen that the two Terms a^ and b^ are contrary to S, they denoting the number of Chances for winning three times together ; it will alfo be feen that the other two Terms -^aab and ^^^bb are partly for him, partly againft him. Let therefore thofe two Terms be divided into their proper parts, viz. "i^aab into cab -\- aba -\- baa, and 'if^bb into abb ■\- bab ■\- bba, and it will plainly be perceived that out of thofe fix parts there are four which are favourable to 5, viz. aab, baa, abb, bba or zaab -\- 2abb ; from whence it follows that the Proljability which S has of winning his Wager will be -- ^ , ^ n — , or dividing both Numerator and Denominator by fi-\-b. it will be found to be ■ _ iL. -, which is the fame as in the preceding Cafe. The reafon of which is, that the winning of a certain number of even Pieces in an odd number of Games is impoffible, un- Igfs it was done in the even number of Games immediately preceding the Ihe Doctrine 0/ Chances. 193 the odd number, no more than an odd number of Pieces can be won in an even number of Games, unlefs it was done in the odd number immediately preceding it ; but ftiil the Problem of winning an even number of Pieces in an odd number of Games is rightly propofed ; for Inflance, the Probability of winning either of one lide or the other, 8 Pieces in 63 Games; for, provided it be done either before or at the Expiration of 62 Games, he who undertakes that it (hall be done in 63 wins his Wager. Case III. Let 2 be the number of Pieces of each Gamefter, and 4 the num- ber of Games upon which the Wager is laid : let therefore a -^ b be raifed to the fourth Power, which is a* A;- /a^a'^b -f- baabb -|- A^ab'^ -|-M; which being done, it is plain that the Terms a-^ -\- /^a^b -\- ^ab'i -j- /^+ are wholly againft S, and that the only Term 6aabb is partly for him, and partly againft him, for which reafon, let this Term be divided into its parts, r/z. aabb, abab, abba, baab, baba, bbaa^ and 4 of thefe parts, "viz. abab, abba, baab, baba, or i^aabb will be found to favour iS ; from which it follows that his Probability of winning will be "■^xji+" • Case IV. If 2 be the number of Pieces of each Gamefter, and 5 the num- ber of Games about which the Wager is laid, the Probability which 5 has of winning his wager will be the fame as in the preceding Cafe, viz. ^aahb Univerfally^ Let 2 be the number of Pieces of each Gamefter, and 2 -\- d the number of Games upon which tlie Wager is laid ; ~2aS^ \-\-~d and the Probability which S has of winning will be . — " - zab — ^ — if ^ be an even number j or -^ ' if d be odd, writing d—\ inftead of d. Case V. If 3 be the number of Pieces of each Gamefter, and 3 -|-</the number of Games upon which the Wager is laid, then the Probabi- C c lity 194 ^^ Doctrine <?/" Chances. lity which S has of winning will be . -,,^ . — i^ d be an 'i,ab i+rf" even number, or t — '• — if it be odd. Case VI. If the number of Pieces of each Gamefler be more than 3, the Expedation of 6\ or the Probability there is that the Play lliall not be ended in a given number of Games, may be determined in the following manner. A General Ride for determinhig ivhat Probability there is that the Play JImU not be determined in a given number of Games. Let n be the number of Pieces of each Gamefter, Let alfo «-|-^ be the number of Games given j raife ^ -|- ^ to the Power «, then cut off the two extream Terms, and multiply the remainder by aa -]- ^ab -|- bb : then cut off again the two Extreams, and multi- ply again the remainder by aa-{- 2ab -\-bb, ftill rejedling the two Extreams ; and fo on, making as many Multiplications as there are Units in —d; make the lafl Product the Numerator of a Fradlion 2 ' whofe Denominator let be a -{- bV'^'^, and that Fradlion willexprefs the Probability required, or the Expedlation of S upon a commorv Stake I, fuppofed to be laid between R and S ; ftill obferving that if ^ be an odd number, you write d — i in its room. Example I. Let 4 be the number of Pieces of each Gamefter, and 10 the number of Games given : in this Cafe « = 4, n -\- dT= lO; where- fore </=6, and -^^=3. Let therefore a-\-b he raifed to the fourth Power, and rejeding continually the extreams, let three Mul- tiplications be made by aa -\- zab -Y bb. Thus, <?+ 1 -^-4^^33-l- baabb-\-^b^\-\-b'^ aa ~\-2ab -{-bb /S^a^b\-\-ba'^bb-\- ^a'b^ -\-%a^bb-\-i2a^b^-\- %aab^ -\- 4a3^3-}- kaab''\\-i,ab^ Mfa''hb T%e Doctrine ^Chances. 195 aa ■\- 2ab -\- bb 4- 14^^!^^+ ioa'^b''\Ar\ifiah ^ 48^5/^5^ 68fl+^*-^ 48^5^5 aa 4- 2<7(^ -4-- ^'^ -i- 48^^^5 468^^^^|-f-48^^^7 \td,a^b''-\-Z'i^2a">b'>^\b\a'^b^ Wherefore the Probability that the Play will not be ended in i o Games will be '+ 2_ ^2^ ■ 4" — ^ which Expreffion will be reduced to -^ , if there be an equality of Skill between the Gamefters ; 10 24 560 %- now this Fradion — — or ~ being fubtrafted from Unity, the 102.). O^ D ^ • remainder will be ~- , which will exprefs the Probability of the Play's ending in 10 Games, and confequcntly it is 35 to 29 that, if two equal Gamefters play together, there will not be four Stakes loft on either fide, in 10 Games. A^. B, The foregoing operation may be very much con traded by omitting the Letters a and b^ and reftoring them after the Lift Mul- tiplication ; which may be done in this manner. Make n -|- -d — i =r^, 2x\di —d •\- \ :=^ q ; then annex to the refpedlive Terms rc- fulting from the laftMuhiplication the literalFrodudsrt/^?, aP~'^ bi+i, at—^-bi+^, &c. Thus in the foregoing Example, inftead of the firft Multiplicand ^a'^b -\- taabb -\- ^ab'^, we might have taken only 4 -|- ^ 4 4. ^'itl inftead of multiplying three times by aa 4 '2-ob 4 '^^j we might have multiplied only by i 4 2 4 i> which would have made the laft Terms to have been 164 4 232 4" ^^A' Now fince that « = 4 and d^= 6, p will be = 6 and ^ = 4, and confequently the literal Produdls to be annexed refpedlively to the Terms 164 4 232 4 164 will be a^b-^, a'~b\ a''b^, which will make the Terms refult- ing from the laft Multiplication to be \(^\a'b'' -\- 22,2a' b^ -\- ib^a'b'', as they had been found before. C c 2 Exam- 196 The Doctrine <?/ Chances. Example II. Let 5 be the number of Pieces of each Gamefter, and 10 the number of Games given : let alfo the proportion of Skill between A and B be as 2 to i. Since « = 5, and ?2 -}- ^ =r 1 o, it follows that d-z=.^. Now d being an odd number muft be fuppofed = 4, fo that -d =z 2 \ let therefore i-}-i be raifed to the fifth Power, and always rejeding the Extreams, multiply twice by 1 4- 2 -^ i , thus ii-1-5-1- '0-1-10+51 + 1 20+35+ 35+ 20 1+2 + 1 1+2+J 5I+10+10-I-5 20H-35+ 35+ 20 -l-IO-]-20 + 20-l-IO +40+ 70+ 70 + 40 + 5+IO+I0I + 5 + 20+ 35 + 35I+2O 20 + 35 + 35 + 20 75+125 + 125 + 75 Now to fupply the literal Products that are wanting, let n + -^ — 1 be made = p, and ^d + i = y, and the Pro- dudls that are to be annexed to the numerical quantities will be aPb'i, aP-^b'i'\-\ ap—^bi-^^, aP—:bi+'^, &c. wherefore;/, in this Cafe, being = 5, and ^=4, then p will be r= 6, and ^ = 3, it follows that the Products to be annexed in this Cafe be a'^b^, a^'b'^y a'^b'^, a^b^^ and confequently the Expedlation of 5 will be found to be N. B. When n is an odd number, as it is in this Cafe, the Ex- pectation of S will always be divifible hy a ■}- b. Wherefore divid- ing both Numerator and Denominator by a + b, the foregoing Ex- preffion will be reduced to ' lT?V ^' 25^333 x—^p^i Let now a be interpreted by 2, and ^ by i, and the Expedation of 5 will become -7—- . PRO- T^e Doctrine ^Chances. ic^j PROBLEM LIX. ^e fa?ne things being given as ift the preceding Problem^ to find the RxpeSiation of Ky or otherivife the Proba- bility that the Play will be ended in a given ntmiber of Games. Solution. Firft^ It is plain that if the Expectation of 5 obtained by the pre- ceding Problem be fubtradled from Unity, there will remain the Expedation of R. Secondly, Since the Expedlation of S decreafes continually, as the- number of Games increafes, and that the Terms we rejefted in the former Problem being divided by aa -\- 2ab + l^l^ are the Decrement of his Expedation; it follows that if thofe reieded Terms he divided continually by aa 4- 2ab-\- bb or a ■\- b\', they will be the Incre- ment of the Expedation of R. Wherefore the Expedation of R may be exprefled by means of thofe rejeded Terms. Thus in the fecond Example of the preceding Problem, the Expedation of R exprefled by means of the rejeded Terms will be found to be yab :o bb " ^ ^ l" a-^b\- \~Z^f^ In like manner, if 6 were the number of the Pieces of each Gamefter, and the number of Games were 14, it would be found that the Expedation of R would be a^-^h^ 1 ''''g , z-aahb ii cab 4.0 And if 7 were the number of Pieces of each Gamefter, and the number of Games were 1 5, then the Expedation of R would be found to be a7-\ -li' ~^o y^tiabb ijj.-.'A> 6;-.+A* N. B. The number of Terms of thefe Series will always be equal to — ^ -|~ ^» if ^ be an even number, or to -~^ , if it br odd. T/jin/l^,. 198 "Tlje Doctrine <?/" Chances. thirdly. All the Terms of thefe Series have to one another cer- tain Relations, which being once difcovered, each Term of any Se- ries relulting irom any Cafe of this Problem, may beeafiiy generated from the preceding ones. Thus in the firft of the two laft foregoing Series, the numerical Coefficient belonging to the Numerator of each Term may be derived from the preceding, in the following manner. Let K, L, M be the three laft Coefficients, and let N be the Coefficient of the next Term required ; then it will be found that N in that Series will con- flantly be equal to 6M — 9L -^ 2K. Wherefore if the Term which would follow • —.rs^ in the Cafe of 16 Games given, were defiredi then make M r= 429, L=iio, K r= 27, and the following Coefficient will be found 1638. From whence it appears that the Term itfelf would be -^^^^j\7r . Likewife, in the fecond of the two foregoing Series, if the Law by which each Term is related to the preceding were demanded, it might thus be found. Let K, L, M be the Coefficients of the three kfi Terms, and N the Coefficient of the Term defired ; then N will in that Series conftantly be equal to 7M — \^'L~\-y\^, or M — 2L -}- K. X 7. Now this Coefficient being obtained, the Term to which it belongs is formed immediately. But if the univerfal Law by which each Coefficient is generated from the preceding be demanded, it will be exprefled as follows. Let Ji be the number of Pieces of each Gamefter : then each Co- jefficient contains n times the laft — n X -—r^ times the laft but one -f- « X •——- X - — ^ times the laft but two — 72 V ''~" X ' "^' ^ X — — times the laft but three 23 + , -,- n X ■ '"~ - X -^— - X "~ X "~'^ limes the laft but four. '-345 &;c. Thus the number of Pieces of each Gamefter being 6, the firft Term n would be = 6, the fecond Term n x —7" would be == 9, the third Term n X ^-^ x -^^^ would be — 2. The reft of the Terms vaniffiing in this Cafe. Wherefore if K, L, M are the three ^ laft "The DocTRiN E o/" Cha NCEs. 199 laft Coefficients, the Coefficient of the following Term will be 6M — 9L+ 2K. Fourthly, The Coefficient of any Term of thefe Series may be found independently from any relation they may have to the pre- ceding : in order to which, it is to be obferved that each Term of thefe Series is proportional to the Probability of the Play's ending in a certain number of Games precifely : thus in the Scries which ex- prelles the Expedtation of R^ when each Gamefter is fuppofed to have 6 Pieces ; ^72;. « + /.> ( lib ~ nf^hb the lafl: Term being multiplied by the common Multiplicator fet down before the Scries, the Produtft d20«* *y<j'^4- » will denote the Probability of the Play's ending in 14 Games pre- cifely. Wherefore if that Term were defired which exprefles the Probability of the Play's ending in 20 Games precifely, or in any number of Games denoted by n -\- d, I fay that the Coefficient of that Term will be - X r-f</-I X »4-y— 5 X «+</— 3 - X '+'/-4. n->rd- 3 4 to fo many Terms as there are Units in —d. — rx-V— x^V- ^-—-^ ^~~T~ to fo many Terms as there are Units in —d — n. 0/ >-fy-. v+d~2 r. +J--, '^+J-^ -r . X 2 >^ 3 ^ 4 ^ 5 to fo many Terms as there are Units in —d — 2«. n+d — I ri+d — 2 n+d — 5 n+d—A V ■ ^ ., _ 3 — -^X X Sec. continued dec. continued 6cc. continued &c. continued 4 S to fo many Terms as there are Units in -d — ■2"- &c. Let now n -\- d he. fuppofed = 20, n being already fuppofed = 6, then the Coefficient demanded will be found from the general Rule to be 6 18 10 ,. iS 17 x-x 16 Id -x — x—x — = 23256 = -18 Wher£- 2 00 T/je Doctrine (?/"Chances. VVlierefbie the Coefficient demanded will be 23256 — 18 = 23238, and then tlie Term itfelf to which this Coefficient does be- long, will be "pnTlT' » ^"^ confequently the Probability of the Play s ending in 20 Games precilely will be ■ — -.^ -x v , — . But feme things are to be obferved about this formation of the Coefficients, which are, Firjf, that whenever it happens that -d, or ~d — n, or jd — 2«, or -(/ — 3;;, &c. expreffing refpedtively the number of Multiplica- tors to be taken in each Line, are = o, then 1 ought to be taken -to liipply that Line. Secondly, That whenever it happens that thofe quantities -d, or J.^/__;;, or -d — 2«, ov-d — 2"' ^^' ^^^ ^^^^ ''^^"^ nothing, other- wife that they are negative, then the Line to which they belong, as well as all the following, ought to be cancelled. PROBLEM LX. Suppojtna- A and B to play together till fuch time as four Stakes are 'won or loft on either ftde ; what tnuft be their proportion of Skilly otherwife nvhat 7nuft be their proportion of ChaJices for winning any ojie Game affigfied, to make it as probable that the Play will be ended in four Ca?nes as not P So LUT I ON. The Probability of the Play's ending in four Games is by the pre- ceding Problem '''^-^^ X i : now becaufe, by Hypothefis, it is to be an equal Chance whether the Play ends or ends not in four Games •, let this Expreffion of the Probability be made = j , then wc fliall have the Equation -^^f = 7 : which, making i>,a:: 1, z, is reduced to "f^^ = T > or .s+ — 42^ — 622; —^z 4. I r= o. Let 122:2; be added on both fides of the Equation, then ^vjll z^ — A.Z'' -\- 6zz — ^z -\- I be = I22;2r, and extrading the Square- the Doctrine of Chances. 201 Square-root on both fides, it will be reduced to this quadratic Equa- tion, zz — 22; -f- I = zV 12, of which the two Roots are X = 5.274 and z ■=. — — — . Wherefore whether the Skill of A be to that of J5, as 5.274 to i, or as i to 5.274, there will be an Equality of Chance for the Play to be ended or not ended in four Games. PROBLEM LXI. Suppojing that A and B play till fuch time as four Staked are won or loji : IVhat muji be their proportion of Skil^ to make it a Wager of three to one^ that the Play will be ended in four Games f Solution. The Probability of the Play's ending in four Games arifing from the number of Games 4, from the number of Stakes 4, and from the proportion of Skill, -viz. of a to ^, is "^j-p^^ ; the fame Pro- bability arifing from the Odds of three to one, is — : Wherefore ' ^+A* " ^^ "' ^^^ fuppofing b, a -.'. i, 2, that Equation will be changed into -^ = J. or 2;* — 122' -|- 3822 — 122 4- i = 5622, and extrading the Square Root on both fides, zz — 62 -|- I :=: z\l ^6, the Roots of which Equation will be found to be 13.407 and ' ■ : ■: Wherefore if the Skill of either be to that of • 3-407 the other as 13.407 to i, 'tis a Wager of three to one, that the Play will be ended in 4 Games. PROBLEM LXII. Suppofing that A and B play till fuch time as four Stakes are won or lofl. What fnufl be their proportion of Skill to make it an equal Wager that the Play will be ended in fix Games f SOLUTTON. Th« Probability of the Play's ending in fix Games, arifing from ths given number of Games 6, from the number of Stakes 4, and D d from 202 The Doctrine t/ Chances. from the proportion of Skill a to b. is — s"" X - — t ' ' ; the fame Probabihty arifing from an equality of Chance, is = — , from whence refults the Equation -=k— v '-p^, ' = — » which ma- king b, a : : 1, z mull be changed into the following z^ -{- 6z^ — 132;* — 2023 — 132:2 -\- 6z -{- 1=0. In this Equation, the Coefficients of the Terms equally diflant from the Extreams, being the fame, let it be fuppofed that the Equa- tion is generated from the Multiplication of two other Equations of the fame nature, viz. zz — yz ■\- i :=.o., and 2* -j- pz^ ■\- qzz -\- pz ■\~ I =z o. Now the Equation refultlng from the Multiplication of thofe two will be Z^ — yz^ 4-12,+ -|- 2/>25 •\-pz -|- ! := o. •\-pz^ — pyz"^ — qyz'^ — yz Arqz'^ which being compared with the firfl: Equation, we fliall have p — y = 6, I — py ■\- q:=z — 13, ip — §^ = — 20, from whence will be deduced a new Equation, '\jiz. y^ -\- 6yy — i6y — 3 2 =r o, of which one of the Roots will be 2.9644, and this being fubftitu- ted in the Equation zz — yz -\- 1 = 0, we (hall at laft come to the Equation zz — 2.96442 + ^ = o> °^ which the two Roots will be 2.C76 and — ^— ; it follows therefore that if the Skill of either Game fter be to that of the other as 2.576 to i, there will be an equal Cfiance for four Stakes to be loft or not to be loft, in fix Games. Corollary If the Coefficients of the extream Terms of an Equation, and likewife the Coefficients of the other Terms equally diftant fronn the Extreams be the fame, that Equation will be reducible to another, in which the Dimenfions of the higheft Term will not exceed half the E)imenfions of the higheft Term in the former. PROBLEM LXIII. Suppojing A and B whofe proportion of Skill is as a io b, fo play together till fuch time as A either wins a cer- tain number q of Stakes^ or B fome other number p of them : what is the Probability that the Play will not be ended in a given number of Games (n)^ Som- The Doctrine (?/"Chances. 203 Solution. Multiply the Binomial a -\-b io many times by it ielf as there arc Units in n — 1, always obferving after every Multiplication to rcjeft thofe Terms in which the Dimenfions of the Quantity a exxeed the Dimenfions of the Quantity b, by 5-; as alfo thofe Terms in which the Dimenfions of the Quantity b exceed the Dimenfions of the Quantity <?, by/»; then fhall the laft Produdl be the Numera- tor of a Fraction exprefling the Probability required, of which Frac- tion the Denominator muft be the Binomial a •\- b raifed to that Power which is denoted by «. Example. Let p be = 3, y = 2, and let the given number of Games be = 7. Let now the following Operation be made according to the foregoing Direftions. a-\-b a-\-b aa\-\-2ab-{'bb _ a-\-b 2aaf^^^2^b^-{-b^ -fit— 2a^b\-\- saabb-^2^b^ a+b Sa'bb-\-%aabi\-\-^ab*- n-yb Sa^bS\-\-Tia'b''-\-%aab^ a\b \'^a''b^-\-z\a^b''\-\-%ab'- From this Operation we may conclude, that the Probability of the Play's not ending in 7 Games is equal to — ^i^.i=^i~- — . Now if an equality of Skill be fuppofed between A and B, the Expref- lion of this Probability will be reduced to • ' ^^l — or — : Where- fore the Probability of the Play's ending in 7 Games will be -^ ; from which it follows that it is 47 to 17 that, in feven Games, either A wins two Stakes of 5, or B wins three Stakes of A. D d 2 PRO- 204- llje Doctrine ^Chances. PROBLEM LXIV. The fame things being fuppofed as in the precedi?ig Problem, to jind the Probability of the Plays enditig in a given number of Games. Solution. Firft, If the Probability of the Play's not ending in the given number of Games, which we may obtain from the preceding Pro- Mem, be fubtraded from Unity, there will remain the Probability of its ending in the fame number of Games. Secondly, This Probability may be exprefled by means of the Terms rejeded in the Operation belonging to the preceding Problem : Thus if the number of Stakes be 3 and 2, the Probability of the Play 'a ending in 7 Games may be exprefled as follows. y\ . ■\ah I 'liaabb X I -1- -^^tT- + Suppofing both a and b equal to Unity, the Sum of the £rft Series will be = -^ , and the Sum of the fecond will be -~ ; which two Sums being added together, the aggregate ~ exprefles the Probability that, in feven Games, either A fhall win two Stakes of 5, or B three Stakes of A. Thirdly, The Probability of the Play's ending in a certain num- ber of Games is always compofed of a double Series, when the Stakes are unequal : which double Series is reduced to a fingle one, in the Cafe of an Equality of Stakes The firft Series always exprefles the Probability there is that A, in a given number of Games, or fooner, may win of B the number q of Stakes, excluding the Probability there is that B before that time may have been in a circumftance of winning the number p of Stakes • both which Probabilities are not inconfiftent together : for A in f ""teen Ganies for Inftance or fooner, may win two Stakes of B, though B before that time may have been in a circumftance of winning three Stakes of A. The fecond Series always expreflfes the Probability there is that B in that given number of Games, may win of yf a certain num- ber "The Doctrine oJ Chances. 205 ber p of Stakes, excluding the Probability there is that A, before that time, may win of jB the number q of Stakes. The firft Terms of each Series may be reprefented refpedtively by the following Terms. aq 1 qab ^ y . ^ -f ^ auho J 9 i?+ < »-!-;• "'* , y . y+i; . ■7 + 6 . <i-\-- . a*b* •^ 1.2.3, "-i""^" ' I . Z . % . 4. . a + b\ ' :^;. ^ I + T^'- "T , . 2 "7+71* » 1.2.3 -^-hA" + . . z . 3 . 4 - "-T^* ' '^^• Each of thefe Series continuing in that regularity till fuch time as there be a number p of Terms taken in the firft, and a number q of Terms taken in the fecond ; after which the Law of the conti- nuation breaks off. Now in order to find any of the Terms following in either of thefe Series, proceed thus: let /> -f ? — 2 be called /; let the CoefRcient of the Term defired be T ; let alfo the Coefficients of the preceding Terms taken in an inverted order, be S, R, Q, P, &c. then will T be equal to /S ^^^ x — -7— R -1 — —— X — ""^ x -~i-Ct-^X^X-^X-~-P, &c. Thusif/>be = 3 and q z=. 1. then / will be 3-J-2 — 2=3, wherefore /S — -- ~' X - ~^ R would in this Cafe be equal to 3S — R, which fhews that the Coefficient of any Term defired would be three times the laft, minus once the lafl but one. To apply this, let it be required to find what Probability there is that in fifteen Games or fooner, either^ fhall win two Stakes of 5, or B three Stakes oi A\ or which is all one, to find what Probability there is that the Play fliall end in fifteen Games at farthert ; A and J5 refolving to play till fuch time as A eitlier wins two Stakes or B three. Let 2 and 3, in the two foregoing Series, be fubf^ituted re pec- tively in the room of q and />, the three firll Terms of the firil Sc-^ ries will be, fetting afide the common Multiplicator, i -f- ~^ — "4" -^ TT^^ ' : likewife the two firfl Terms of the fecond will be I ^ —rp*" - ^^'^^ becaufe the Coefficient of any Term defired in each 2o6 Tlie Doctrine ^Chances. each Series is refpedtively three times the laft, minm once the laft but one, it follows that the next Coefficient in the firfl Series will be found to be 13, and by the fame Rule the next to it 34, and fo on. In the fame manner, the next Coefficient in the fecond Series will be found to be 8, and the next to it 21, and io on. Wherefore rcftoring the common Multiplicators the two Series will be If we fuppofe an equality of Skill between A and 5, the Sum of the firfl Series will be — r|g- » the Sum of the fecond will be and the Aggregate of thofe two Sums will be 3,-5S ' "^ 00 o .,276s ' which will exprefs the Probability of the Play's ending in fifteen Games or fooner. This laft Fradtion being fubtraded from Unity, there will remain - '^^"1 , which expreffes the Probability of the Play's 3.' 7. ',8 * •' continuing beyond fifteen Games: Wherefore 'tis 31 171 to 1597, or 59 to 2 nearly that one of the two equal Gamefters that fhall be pitched upon, (hall in fifteen Games at fartheft, either win two Stakes of his Adverfary, or lofe three to him. N. B. The Index of the Denominator in the laft Term of each Series, and the Index of the common Multiplicator prefixed to it being added together, muft either equal the number of Games given, or be lefs than it by Unity. Thus in the firft Series, the Index 12 of the Denominator of the laft Term, and the Index 2 of the common Multiplicator being added together, the Sum is 14, which is lefs by Unity than the number of Games given. So like- wife in the fecond Series, the Index 1 2 of the Denominator of the laft Term, and the Index 3 of the common Multiplicator being added together, the Sum is 15, which precifely equals the number of Games given. It is carefully to be obfervcd that thofe two Series taken together exprefs the Expedlation of one and the fame perfon, and not of two different perfons ; that is properly of a Spedator, who lays a wager that TIdb Doctrine <?/ Chances. 207 that the Play will be ended in a given number of Games. Yet in one Cafe, they may exprefs the Expedlations of two different perfons : for Inftance, of the Gamefters themfelves, provided that both Series be continued infinitely; for in that Cafe, the firfl Series infinitely continued will exprefs the Probability that the Gamefter A may fooner win two Stakes of 5, than that he may lofe three to him : likewife the fecond Series infinitely continued will exprefs the Pro- bability that the Gamefler B may fooner win three Stakes of y^, than lofe two to him. And it will be found, (when I come to treat of the Method of fumming up this fort of Series, whofe Terms have a perpetual recurrency of relation to a fixed number of prece- ding Terms) that the firfl Series infinitely continued is to the fecond infinitely continued, in the proportion of aa x aa ■\- ab ■\- bb to b"" x a -\- b; that is in the Cafe of an Equality of Skill as 3 to 2, which is conformable to what I have faid in the ix'*" Problem. Fourthly, Any Term of thefe Series may be found independently from any of the preceding : for if a Wager be laid that A fliall either win a certain number of Stakes denominated by q, or that B fhall win a certain number of them denominated by/>, and that the number of Games be exprefTed by 5- -|- ^; then I fay that the Coefficient of any Term in the firft Series anfwering to that number of Games will be "1 I ^ 2 X X 3 " 4 many Multiplicatorsas there are Units in -d. &c. continued to fo --4^x q+d—\ q+d- X ?+-^— 3 &c. continued to fo many Terms as there are Units in -;d—p. &c. continued to fo- many Terms as there are Units in ~d—p—q. 37+^/> q+d- X 1^-d- 4 continued to fo many Terms as there are Units in ^d — 2p — q. + X X X .3 •■ 4 many Terms as there are Units in ^d—2p — zq, ~1 ^ ~ 4 &c. continued to fo X &c. continued to fb many Terms as there are Units, in ^d^^p—zq. -V 79+ ^f 2o8 "The Doctrine o/" Chances. H ■ • X -^— ; X , ■ X ~ , &c. continued to fo many Terms as there are Units in L^^^p—'^q. And fo on. And the fame Law will hold for the other Series, calling /)-f«J the number of Games given, and changing q into p, and p into a, as alfo d into ^, ftill remembring that when d is an odd number, a — i ought to be taken in the room of it, and the like for S. And the fame obfervation muft be made here as was made at the end of the lix'*" Problem, viz. that \i -^d, ox-d—p, or -d — p — y, or ' d — 2/) — q, or —d — 2p — zq^ &c. exprefllng refpeftively the number of Multiplicators to be taken in each Line, are — o, then 1 ought to be taken for that Line, and alfo, that if ~d,or^d — p, or —d — p — q, Sec. arc lefs than nothing, other- wife negative, then the Line to which they belong as well as all the following ought to be cancelled. PROBLEM LXV. Jf A a^id B, whofe proportion ofjkill is fuppofed as a to b, play together : JVhat is the Probability that one of them, Juppofe A, may i?t a number of Games not ex- ceeding a nufnber given, win of ^ a certain number of Stakes f leaviyjg it wholly indifferent whether B, before the expiration of thofe Games, may or may not have been in a circumflance of winning the fa?ne, or any other number of Stakes of K. Solution. Suppofing n to be the number of Stakes which A is to win of 5, and n ^\- d the number of Games ; let a-\-bht raifed to the Power whofe Index is n-\'d -, then if d be an*odd number, take fo many Terms of that Power as there are Units in — ^ ; take alfo fo many of the Terms next following as have been taken already, but prefix to them in an inverted order, the Coefficients of the preceding Terms. But if </ be an even number, take fo many Terms of the faid Ihe Doctrine of Chances. 209 faid Power as there are Units in -d-\- 1 ; then take as many of the Terms next following as there are Units in 4^> and prefix to them in an inverted order the Coefficients of the preceding Terms, omitting the lift of them ; and thofe Terms taken all together will compofe the Numerator of a Fradlion exprefling the Probability re- quired, the Denominator of which Fradion ought to be a -\- ^V+'^- Example I. Suppofing the number of Stakes, which A is to win, to be Three, and the given number of Games to be Ten ; \ti a -\- bhc raifed to the tenth power, mz, a'° -\- loa'^b -\- \^abb ■\- izoa'^b^ -\- zioa^b^ -\-2.^za^b'^ -\- 2ioa'^b^ ■\- izca'^b'' -^^^aap -\- lOdb^ -{- b'°. Then by reafon that « = 3, and n -\- d =z 10, it follows that </ is = 7, and "^' =: 4. Wherefore let the Four firft Terms of the faid Power be taken, viz. a^° -\- loa'^b ■\- ^^a^bb -\- \zoalb\ and let the four Terms next following be taken likewife without re- gard to their Coefficients, then prefix to them in an inverted order, the Coefficients of the preceding Terms : thus the four Term.s fol- lowing with their new Coefficients will be izoa^b"^ -^ \K^a'->b'> -\- joa^b^ -\- la^bT. Then the Probability which A has of winning three Stakes of B in ten Games or fooner, will be exprefTed by the following Fraction a^''-ir\oa1b-ir\'^a%h-h-\i^a'h"'-\-\zc_aH^.\-j^^a^li-\-\ca*b^-'ra^b'' which in the Cafe of an Equality of Skill between A and B will be reduced to -^^ or — . 1024 32 Example II. Suppofing the number of Stakes which A has to win to be Four, and the given number of Games to be Ten ; let a -]- b be raifed to the tenth Power, and by reafon that n is =r 4, and « -|- ^ = i o, it follows that ^ is = 6, and \d A^ i = 4 j wherefore let the four lirfl: Terms of the faid Power be taken, viz. a'° -\- loa'^b -f- /\.^a^bb -f- 120^7/^5 . take alfo three of the Terms following, but prefix to them, in an inverted order, the Coefficients of the Terms already taken, omitting the laft of them; hence the three Terms following with their new Coefficients will be /:^^a^b^-\- loa^b'' -{• la^b^. Then E e the 2IO He Doctrine ^Chances. the Probability which A has of winning four Stakes of JS in ten Games, or fooner, will be exprefTed by the following Fradion which in the Cafe of an Equality of Skill between A and B will be reduced to — — or — -r . 1024 120 Another Solution. Suppofing as before that « be the number of Stakes which A is to win, and that the number of Games be n -|- d, the Probability which A has of winning will be exprefled by the following Series -|- ^ V ' "T~T^r^'~^ — > ^^- which Series ought to be con- tinued to fo many Terms as there are Units in ^d-\-i ; always ob- ferving to fubftitute d — i in the room of d in Cafe ^be an odd num- ber, or which is the fame thing, taking fo many Terms as there are Units in . Now fuppofing, as in the firil Example of the preceding Solution, that Three is the number of Stakes, and Ten the given number of Games, and alfo that there is an equality o f Skill between A and B, the foregoing Series will become jX i + — -{ — \ -j — — — • — , as before. Remark. In the firft attempt that 1 had ever made towards folving the ge- neral Problem of the Duration of Play, which was in the Year 1708, I began with the Solution of this lxv'** Problem, well knowing that it might be a Foundation for what I farther wanted, fince which time, by a due repetition of it, I folved the main Problem : but as I found afterwards a nearer way to it, I barely publifhed in my firft Effay on thofe matters, what feemed to me moft fimple and elegant, ftill preferving this Problem by me in or- der to be publilhed when I fhould think it proper. Now in the year 17 13 Mr. de Monmort printed a Solution of it in a Book by him publiflied upon Chance, in which was alfo inferted a Solution of the fame by Mr. Nicolas Bernoulli; and as thofe two Solutions feemed to Ihe Doctrine(?/"Chances. 211 to me, at firft fight, to have feme affinity with what I had found before, 1 confidered them with very great attention ; but the Solu- tion of Mr. Nicolas BermuHi being very much crouded with Symbols, and the verbal Explication of them too fcanty, I own I did nor underftand it thoroughly, which obliged me to confider Mr. de Mon- mort\ Solution with very great attention : I found indeed that he was very plain, but to my great furprize I found him very erroneous ; ftill in my Dodlrine of Chances I printed that Solution, but rediiied and afcribed it to Mr. de Monfnort, without the lead intimation of any alterations made by me ; but as I had no thanks for fo doing, I refume my right, and now print it as my own : but to come to the Solution. Let it be propofed to find the number of Chances there are for A to win two Stakes of B, or for B to win three Stakes of ^, in fifteen Games. The number of Chances required is exprefled by two Branches of Series ; all the Series of the firft Branch taken together exprefs the number of Chances there are for A to win t .vo Stakes of B, exclufive of the number of Chances there are for B before that time, to win three Stakes of A. All the Series of the fecond Branch taken together exprefs the number of Chances there are for B to win three Stakes of ^, exclufive of the number of Chances there are for A before that time to win two Stakes of B. Firjl Branch of Series. flU a^*b a'ihi a'^i! a"i* a'°&s a^i^ a^hl ah'' aHo a^h" a^i" aii"- aH'i 1 + 15 + 105 + 4554-1365 + 3003 + 5005 + 5005 + 3005+1365 + 455+ 105+ 15 +1 — I — «5 — «05— 455— 455 — '03 - 15 — ' + I + 15 + 15 + I Second Branch of Series. 4'S iHa b'^ia^ l>"a* b'°a^ b^a'' b^di bU^ b^tfl bU'" b^a" b^a" i'a'« 1+15 + 105 +455 +1365 + 3003 +5005 + 3003 + 135.5 + 455+105+ 15 +1 — « — 15 — 105 - 455 —1365—45; — '05 — 15 — 1 + I + 15 + I The literal Quantities which are commonly annexed to the nu- merical ones, are here written on the top of them ; which is done, to the end that each Series being contained in one Line, the de- pendency they have upon one another, may thereby be made more confpicuous. E e 2 The 2 12 T'he Doctrine i?/' Chances. The firfl Series of the firft Branch expreffes the number of Chan- ces there are for A to win two Staines of B, including the number of Chances there are for B before, or at the Expiration of the fifteen Games, to be in a Circumftance of winning three Stalces of A ; which number of Chances may be deduced from the lxv'"' Pro- blem. The fecond Series of the firft Branch is a part of the firft, and expreffes the number of Chances there are for B to win three Stakes of A, out of the number of Chances there are for A^ in the firfl: Series to win two Stakes of B. It is to be obferved about this Se- ries, Firjiy that the Chances of B expreffed by it are not reftirained to happen in any order, that is, either before or after A has won two Stakes of B. Secondly, that the literal produd^s belonging to it are the fame with thofe of the correfponding Terms of the firft Series. Thirdly, that it begins and ends at a'n Interval fi-om the firft and laft Terms of the firft Series equal to the number of Stakes which B is to win. Fourthly, that the numbers belonging to it are the numbers of the firft Series repeated in order, and continued to one half of its Terms ; after which thofe numbers return in an inverted order to the end of that Series : which is to be underftood in cafe the number of its Terms fliould happen to be even ; for if it fliould happen to be odd, then that order is to be continued to the greateft half, after which the return is made by omitting the laft number. Fifthly, that all the Terms of it are affedcd with the fign ininus. The Third Series is part of the fecond, and expreffes the num- ber of Chances there are for A to win two Stakes of B, out of the number of Chances there are in the fecond Series for B to win three Stakes of A -, with this difference, that it begins and ends at an Interval from the firft and laft Terms of the fecond Series, equal to the number of Stakes which A is to win ; and that the Terms of it are all pofitive. It is to be obferved, that let the number of thofe Series be what it will, the Interval between the beginning of the firft and the beginning of the fecond, is to be equal to the number of Stakes which B Is to win ; and that the Interval between the beginning of the fe- cond and the beginning of the third, is to be equal to the number of Stakes which A is to win ; and that thefe Intervals recur alter- nately in the fame order. It is to be obferved likewife that all thefe Scries are alternately pofitive and negative. All the Obfervations made upon the firft Branch of Series be- longing alfo to the fecond, it would be needlefs to fay any thing more of them. Now Ithe Doctrine <?/'Chances. 213 Now the Sum of all the Series of the firft Branch, being added to the Sum of all the Series of the fecond, the Aggregate of thefe Sums will be the Numerator of a Fraftion exprefling the Proba- bility of the Play's terminating in the given number of Games; of which the Denominator is the Binomial a Ar b raifed to a Power whofe Index is equal to that number of Games. Thus fuppofing that in the Cafe of this Problem both a and b are equal to Unity, the Sum of the Series in the firft Branch will be 18778, the Sum of the Series in the fecond will be 12393, and the Aggregate of both 31 17 I ; and the Fifteenth Power of 2 being 3276^, it follows that the Probability of the Play's terminating in Fifteen Games will be ^.^g , which being fubtrafted from Unity, the remainder will be - ^]^l^ - '• From whence we may conclude that it is a Wager of 3 1 17 1 to 1597, that either A in Fifteen Games (hall win two Stakes of .8, or B win three Stakes of A : which is conformable to what was found in the lxiv'*' Problem. PROBLEM LXVI. 'To jind That Probability there is that ifi a given number of Games A may be wiiiner of a certain number q of Stakes, and at fome other time B may likewife be winner of the number p of Stakes^ fo that both cir^ cumfiances may happen. Solution. Find by onr lxv'^ Problem the Probability which A has of winr- jiing, without any limitation, the number q of Stakes : find alfa by the ixiii'' Problem tlie Probability which A has of winning that number of Stakes before B may happen to win the number p ; then from the firft Probability fubtrading the fecond, the re- mainder will exprefs the Probability there is that both A and B may be in a circumftance of winning, but B before A. In the like man- ner, from the Probability which B has of winning without limita- tion, fubtrading the Probability which he has of winning before A, the remainder will exprefs the Probability there is that both A and B may be in a circumftance of winning, but A before B: wherefore adding thefe two remainders together, their Sum will ex- prefs the Probability required. Thus; 214 ^^ Doctrine o/" Chances. Thus if it were required to find what Probability there is, that in Ten Games A may win Two Stakes of By and that at fome other time B may win Three : The firft Scries will be found to be The fecond Series will be The difference or thele Series beine -=77^^ X — r->6 + " '-TrvT D ti-\-ti'- a-\r\\° I «-t-t>° expreffes the firft part of the Probability required, which in the Cafe of an equality of Skill between the Gameflers would be reduced to 3 The third Series is as follows, ^ab tjaabb 2'aa^b^ The fourth Series is a-\-b 6i 3«o --uabb 7^ 2Ia''' 6 /,^ aabb ya^b^ The difference of thefe two Series being -^qr^r X -— r^*'-!- - ^^6 expreffes the fecond part of the Probability required, which in the Cafe of an equality of Skill would be reduced to -^ . Wherefore the Probability required would in this Cafe be -j-:^ 4" "777" ="777" • Whence it follows, that it is a Wager of 495 to 17, or 29 to i very near, that in Ten Games A and 5 will not both be in a circumftance of winning, viz. A the number q and B the number p of Stakes. But if by the conditions of the Problem, it were left indifferent whether A or B fhould win the two Stakes or the three, then the Probability required would be increafed, and become as follows j •viz. fl-i-l-il aabb ■ja'b^ •T" ^+2^3 X -^ip^^ 1 ^+A<> which 'The Doctrine of Chances. 215 which, in the Cafe of an equality of Skill between the Gamefters, would be double to what it was before. PROBLEM LXVII. 72> find what Probability there is^ that i?i a given nu7n- ber of Games A may win the 7iumber q of Stakes ; with this farther condition^ that B during that whole number of Games may never have kee^i winner of the number p of Stakes. Solution, From the Probability which A has of winning without any li- mitation the number q of Stakes, fubtradl the Probability there is that both A and jB may be winners, viz. A of the number q, and B of the number p of Stakes, and there will remain the Probability required. But if the conditions of the Problem were extended to this alter- native, viz. that either A fliould win the number q of Stakes, and B be excluded the winning of the number p j or that B (hould win the number^ of Stakes, and y^ be excluded the winning of the num- ber ^, the Probability that either the one or the other of thefe two Cafes may happen, will eafily be deduced from what we have faid. The Rules hitherto given for the Solution of Problems relating to the Duration of Flay are eafily pradlicable, if the number of Games given is but fmall ; but if that number is large, the work will be very tedious, and fometimes fwell to that degree as to be in fome manner imprafticable : to remedy which inconveniency, I fhall here give an Extradt of a paper by me produced before the Royal Society, wherein was contained a Method of folving very expeditioufly the chief Problems relating to that matter, by the help of a Table of Sines, of which I had before given a hint in the firft Edition of my JDo^r/wd- o/'CZwwff^, pag. 149, and 150. PROBLEM LXVIIL To fohe by a Method different from any of the pre- ceding., the Problem lix, when a is to b in a ratio of Equality. SOLU- 2 1 6 The Doctrine <?/" Chances. Solution. Let 71 be the number of Games given, and p the number of Stakes ; let Q reprefent 90 degrees of a Circle whofc Radius is equal to Unity ; let C, D, E, F, &c. be the Sines of the Arcs iL J^ ^LL J:S. &c. till the Quadrant be exhaufted ; let aUb, c, d, e, f, Sec. be the Co-fines of thofe Arcs : then if the difference between a and p be an even number, the Probability of the Play's not ending in the given number of Games will be repre- fented by the Series »4-i ,"+1 ,»-}-i ^n+l of which Series very few Terms will be fufficient for a very near approximation. But if the difference between n and p be odd, then z c" d" " f" the Probabity required will be-X-;- ir+"i: F" &c. In working by Logarithms, you are perpetually to fubtradt, from the Logarithm of every Term, the Produdl of 10 into the number «, in cafe the number « — p be even ; but in cafe it be odd, you are to fubtrad the Produd of 10 into n — i, and if the Subtrac- tion cannot be made without making the remainder negative, add 10, 20, or 30, &c. and make fuch proper allowances for thofc addi- tions as thofe who are converfant with Logarithms know how to make. To apply this to fome particular cafes, let it be required to find the Probability of Twelve Stakes being not loft in 108 Games. Here becaufe the difference between 108 and 12 is 96, I take the firft form, thus The Arcs -y , -j- , -y- , —p , -y , —j- . —j- > «c. being refpedively 7^—3°') za'' — 30', 37^ — 3°'' 5^'^— 3°'. (fjj qo', 82'' — 30', gy^ — 30', &c. I take only the fix firft, as not exceeding 90'' Now the Logarithm of the Co-fine of y'^ — 30' being 9.9962686, I multiply it by 7i -f- I, that is in this Cafe by 109, and the produdl will be 1089.5932774, which is the Logarithm of the Numerator of the firft Fradion — ^ — . From "The Doctrine o/" Chances. 217 From that Logarithm, I fubtradl the Logarithm of the Sine of 7</ — 20' here reprefented by C, which being 9. 11 56977, the re- mainder will be 1080.4775797, out of which rejedting loSo pro- dudl of 10 by the given number of Games 108, and taking only 0.4775797 the number anfwcring will be 3.00327, which being multiplied by the common Multiplicator — , that is in this Cafe by — or -^ , the produdt will be 0.50053, which Term alone de- termines nearly the Probability required. For if we intend to make a Corredion by means of the fecond Term — j3 — , we fhall find the Logarithm of — j^— to be 1076.6692280 to which adding 10, and afterwards fubtrading 1080, the remain- der will be 6.6692280, to which anfwers 0.0004669, of which the 6^'' part is 0.0000778, which being almofl nothing may be fafely rejedled. And whenever it happens that « is a large number in refped to />, the firft Term alone of thefe Series will exceeding near determine the Probability required. Let it now be required to find the Probability of 45 Stakes being not loft on either fide in 1519 Games. The Arcs -y-- , -^ , -^ , &c. being refpedively 2^ 6^, 10^^ &c. I take, 1°, the Logarithm of the Co-fine of 2^ which is 9.9997354, which being multiplied by n -\- i, that is in this Cafe by 1520, the produdl will be 1 5199.5988080, out of which fub- trafting the Logarithm of the Sine of 2', viz.. 8.5428192, the re- mainder will be 15191.0559S88, out of which rejecting 15190, the number anfwering will be 1 1.3759, which being multiplied by -^, that is, in this Cafe by -^ , the prouu6t will be .50559 which nearly determines the Probability required. Now if we want a Corredion by means of the fecond Term we fhall find -^-j^; — n:= .00002081, which Term being fo very inconfiderable may be entirely rejeded, and much more all the following. Confidering therefore that when the Arc — is fmall the firft Term alone is fufficient for a near approximation, it will not be amifs to inquire what muft be the number of Games that fliall make it an equal Probability of the Play's being ended in that num- ber of Games J which to do, F f Suppofe 21 8 The Doctrine 6/" Chances. Suppofe —^ — X - = 7, hence 4^+' = C/*, then fup- pofing p a large number, whereby the number n muft be ftill much larger, we may barely take for our Equation 4^ = />C, then taking the Logarithms, we fliall have Log. 4 + « Log. c = Log. C -i- Log. /), let the magnitude of the Arc — be fuppofed = z ; now fince the number p has been fuppofed very large, it follows that the Arc s muft be very fmall ; wherefore the Sine of that Arc will alfo be nearly = z, and its Co-fine i ^-zz nearly, of which Co-fine the Logarithm will be — ^zz nearly ; we have therefore the Equation Log. 4 — ^nzz. = Log. p -j- Log. z ; let now the Mag- nitude of an Arc of 90'', to a Radius equal to Unity, be = M, hence we fhall have — — = 2;, and Log. z ^=. Log. M — Log. p^ wherefore the Equation will at laft be changed into this, Log. 4 >_ i!^= Log. M, and therefore n = ^ '"-■ \- ; '°'^- ^ . X/-/. ^^^ ^'^s-4— j'pg- M __ 0.756 nearly, and therefore n = 0.756//'. N. B. The Logarithms here made ufe of are fuppofed to be Hy- perbolic Logarithms, of which I hear a Table will foon be pub- liflied. ■ Mr. (fe Monmort in the fecond Edition of his Tracfl, Des jeux de Hazard^ tells us that he found that if p denoted an odd number of Stakes to be won or loft, making —^ z=zj, that then the Quan- tity 'iff— 3/-I- I would denote a number of Games wherein there would be more than an equal Probability of the Play's being ended ; but at the fame time he ov/ns, that he has not been able to find a Rule like it for an even number of Stakes. Whereupon I fliall obferve, Jirjl, that his Exprefilon may be reduced to ~pp -\- ~ . Which tho' near the Truth in fmall numbers, yet is very defedive in large ones, for it may be proved that the number of Games found by his Exprelfion, fat from being above what is requifite, is really below it. Secondly, that his Rule does not err more in an even number of Stakes than in an odd one ; but that Rule being founded upon an indudion gathered from the Solution of feme of the fimpleft Cafes of this Problem, it is no wonder that he reftrained it to the odd Cafes, he happening to be miftaken in deter- mining The Doctrine ^Chances. 219 mining the number of Games requifite to make it an even Wager that twelve Stakes would be won or loft before or at the expiration of thofe Games, which he finds by a very laborious calculation to have been 122; in which however he was afterwards redified by Mr. Nicolas Bernoulli, who informed him that he had found by his own Calculation that the number of Games requifite for that purpofe was above 108, and below iio; and this is exaftly conformable to our Rule, for multiplying pp -7= 144 by 0.756, the Produ(ft will be 108.864. For a Proof that his Rule falls fhort of the Truth, let us fuppofe p = 45, then/will be =: 23, and ^Jjf — 3/4" ^ will be = 1519, let us therefore find the Probability of the Play's terminating in that number of Games; but we have found by th'S lxviii'*' Pro- blem, that the Probability of the Play's not terminating in that number of Games is 0.50559 ; and therefore the Probability of its terminating within them is 0.49441; which being lefs than — , {hews 'tis not more than an equal Wager that the Play would be terminated in 151 9 Games. But farther, let us fee what number of Games would be neceffary for the equal wager, then multiplying 2025 fquare of 45 by 0.756, the Product will be 1530.9 ; which fhews that about 1531 Games are requifite for it. PROBLEM LXIX. 'I%e Jame things being given as in the preceding Problem, except that now the ratio of 3. t oh is fuppofe d of inequa- lity, to fohe the fame by the Si?2es of Arcs. Solution. Let n reprefent the number of Games given, /; the number of Stakes to be won or loft on either fide, let alfo A be the Semi- circumference of a Circle whofe Radius is equal to Unity : let C, D, E, F, &c. be the Sines of the Arcs — , — - , -^^^— , — , &c. till the Semi-circumference be exhaufted ; let alfo c, d, e,/, &c. be the refpedive verfed Sines of thofe Arcs ; let - . . ^^ be made = L, ri" ^' :=/, ^-=r; letc, 2r : : CC, w; d, zr :: DD, ^r; F f 2 e. 2 20 T*Ih Doctrine o/'Chances. (f, 2r : : EE, f-, &c. then the Probability of the Play not ending in « Games will be exprefled by the following Series ^ T the whole to be multiplied by Lp-r p\r As there are but few Tables of Sines, wherein the Logarithms of the verfed Sines are to be found, it will be eafy to remedy that in- convenlency, by adding the Logarithm of 2 to the excefs of twice the tabular Logarithm of the Sine of half the given Arc above 10 ; for that Sum will give the Logarithm of the verfed Sine of the whole Arc. It will be eafily perceived that inftead of referring the Arcs to the Divifion of the Semi-circumference, we might have referred them to the Divifion of the Quadrant, as in the Cafe of the preceding Problem. Of the Summation of recurring Series. The Reader may have perceived that the Solution of feveral Pro- blems relating to Chance depends upon the Summation of Series ; I have, as occafion has offered, given the Method of fumming them up ; but as there are others that ma.y occur, I think, it neceffary to give a fummary View of what is moft requifite to be known in this matter ; defiring the Reader to excufe me, if I do not give the De- monftrations, which would fwell this Trad: too much ; efpecially conlidering that I have already given them in my Mlfcellanea Ana- lytica. I call that a recurring Series which is io conftituted, that having taken at pleafure any number of its Terms, each following Term fhall be related to the fame number of preceding Terms, according to a conftant law of Relation, fuch as the following Series A B C D E F In which the Terms being re/pedively reprefented by the Capitals A, B, C, D, &c. we fhall have D = 3CAr — zExx ■\- fA.v^ E = 3 Dx — zCxx -\- ^^x^ F = 3EA' — zDxx -j- 5Cx^ &c.. Now The Doctrine o/" Chances. 221 Now the Quantities 3^ — 2xx-\-t^x^, taken together and con- nedled with their proper Signs, is what I call the Index, or the Scale of Relation ; and fometimes the bare Coefficients 3 — 2 -|- 5 are called the Scale of Relation. Proposit ion I. If there be a recurring Series a -\- bx -\- cxx -\- dx"^ -\- ex*, &c. of which the Scale of Relation bey3i: — gxx ; the Sum of that Series continued i?2 injinitum will be a-\-bx —fox i — fo-^gx>i Proposition II. Suppofing that in the Series a -|- bx -\- cxx •\- dx^ -\- ex*, &c. the Law of Relation he fx — gxx ■\- hx^ ; the Sum of that Series continued in infinitum will be a •\- bx -|- cxx — fax — Jbxx -]- gaxx I — fx 4" gxx — hx''' Proposition III. Suppofing that in the Scries, a -{- bx -|- cxx, &c. the Law of Re- lation hejx — gxx -\- hx'" — kx*, the Sum of the Series will be a-\- bx -j- cxx -|- dx^ — fox — fbxx — fcx^ -{- gaxx -\- gbx'^ — bax'i I — fx -\- gxx — bx'^ -\- kx* As the Regularity of thofe Sums is confpicuous, it would be need- lefs to carry them any farther. Still it is convenient to know that the Relation being given, it will be cafy to obtain the Sum by obferving this general Rule. 1 °, Take as many Terms of the Series as there are parts in the Scale of Relation. 2°, Subtrad: the Scale of Relation from Unity, and let the re- mainder be called the Differential Scale. 1%, MuL- 22 2 7/;^ Doctrine (t/* Chances. 3°, Multiply thofe Terms which have been taken in the Series by the Differential Scale, beginning at Unity, and fo proceeding or- derly, remembering to leave out what would naturally be 'extended beyond the laft of the Terms taken. Then the Produdl will be the Numerator of a Fradion exprefling the Sum, of which the Denominator will be the Differential Scale. Thus to form the preceding Theorem, Multiply a -\-l>x ■]- cxx -\- dx'^ ^y I — f^ -\'' S^^ — '^■^' • • • and beginning from Unity, we (hall have a-\- bx -\- cxx -\- dx'^ — fax — Jbxx — fcx"^ ... -L- gaxx -\- gbx^ . . . — hax'^ . . . omitting the fuperfluous Terms, and thus will the Numerator be formed ; but the Denominator will be the Differential Scale, viz. 1 — Jx -\- gxx — •bx'i -\- kx^. Corollary. If the firft Terms of the Series arc not taken at pleafure, but be- gin from the fecond Term to follow the Law of Relation, in fo * much that b fhall he:=fa c —fb—ga d ^=^fr — gb ■\- ha &c. then the Fradion expreffmg the Sum of the Series will have barely the firil Term of the Series for its Numerator. Proposition IV. If a Series is fo conftituted, as that the laft Differences of the Coefficients of the Terms whereof it is compofed be all equal to no- thing, the Law of the Relation will be found in the Binomial 1 — x^ " , n denoting the rank of thofe laft Differences ; thus fup- pofing the Series A B C D E F G \-\- ^x ■\- loxx + zox'^ + SS'^* +■ 56.V5 -f- 84^^ &c. whereof the Coefficients are, I -1-4 'Jthe Doctrine <?/■ Chances. 223 14-4+10 4-20 -i-35 +56 -I- 84 i*"' Differences - - 3 + 6 +10 +15 -j- 21 4-28 2'' Differences 3 +4 -i-5 +6 -I-7 3'' Differences i-f"i "hi "("' 4''' Differences - — o +0 +0 I fay that the Relation of the Terms will be found in the Binomial I — x^ *, which being expanded will be i — \x -j- bxx — \x'^ -|~x+ and is the Differential Scale, and therefore the Scale properly fo cal- led will be 4X — bxx -|- 4x3 — x'' j thus, in the foregoing Series, the Term G r=r 4FX — 6EArx 4" 4^x5 — iCx''-. Corollary. The Sums of thofe infinite Series which begin at Unity, and have their Coefficients the figurate numbers of any order, are always expreffible by the Fradion ^. , wherein p denotes the rank or order which thofe figurative numbers obtain ; for Inftanc>.- if we take the Series I -f- IX -|- ixx Ar \x^ -j- IX''- -j- i>^' -1- i^^, &c. which is a geo- metric Progreffion, and whofe Coefficients are the numbers of the firft order, the Sum will be — ^ — , and if we take the Series I X I -]- 2x -}- Z^^ + 4'^'^ -\- S^^ + 6a'5 + 7^^ &c. whofe Coefficients compofe the numbers of the fecond order, the Sum will be v, • and again, if we take the Series i +3x-|- 6xx -}- Jox^ -j- i5x% &c, whofe Coefficients are the numbers of the third order, other- wife called Triangular numbers, the Sum will be "^=^7^" • Proposition V. The Sum of any finite number of Terms of a recurring Series a -[- bx -j- rxx -f- dx^ -]- ^x''^, &c. is always to be obtained. Thus fuppofing the Scale of Relation to be fx — ^xx ; 71 the number of Terms whofe Sum is required; and ax' -j' /3x''+' the two Terms which would next follow the laft of tlie given Terms, if the Series was continued ; then the Sum will be a -^-bx — x" X a -|- /3x — fax — /ax 1 — y3c _|- ^xx But 2 24 ^^ Doctrine <?/" Chances. But if the Scale of Relation be fx — gxx -f- hx^, n the number of Terms given, and ax'' -j- /3\" + ' 4-y^''"'"S the three Terms that would next follow the kft of the given Terms, then the Sum will be a ^bx -\- cxx — X" X « -{- jGa: -|- yxx — fax — fbxx — jocx — //3x -|- gaxx ^ga-xx \ —. fx -\- gxx — hx^ The continuation of which being obvious, thofe Theorems need not be carried any farther. But as there is a particular elegancy for the Sums of a finite num- ber of Terms in thofe Series v/hofe Coefficients are figurate num- bers beginning at Unity, I fliall fet down the Canon for thofe Sums. Let n denote the number of Terms whofe Sum is to be found, and p the rank or order which thofe figurate numbers obtain, then the Sum will be » , n vx w . w-f-I . .r . »+ 2 . — I n X 1 . z .7^7)?-- n . »+ 1 . n-\- 2 . » 4- 3 , &C. 3 .T^rP^/' — 3 1.2.3 4 • I— ^^-4 which is to be continued till the number of Terms be =p. Thus fuppofing that the Sum of twelve Terms of the Series, I -\- 3X -\- txx -\- 10x3 -|- i^x*, &c. were demanded, that Sum will be ~rrz\^ Proposition VI. In a recurring Series, any Term may be obtained whofe place is affigned. It is very plain, from what we have fuid, that after having taken fo many Terms of the Series as there is in the Scale of Relation, the Series may be protrafted till it reach the place affigned ; however if that place be very dirtant from the beginning of the Scries, the • continuation of thofe Terms may prove laborious, efpecially if there be many parts in the Scale. But there being frequent Cafes wherein that inconveniency may be avoided, it will be proper to fhew by what Rule this may be known ; and then to fliew how we are to proceed. The Rule'will be to take the Differential Scale, and to fuppofe it — o, then if the roots of that fuppofed Equation be all real, and unequal, the thing may be effedted as follows. Let the Series be re- prefented by ^ + ^'" 7he Doctrine c/Chai\-ces. 225 aArbr -\-crr -j- dr^ -j- er*, &c. and 1° i(fr — grr be the Scale of Relation, and confequently i — ff -y E^^ the differential Scale, then having made i — Jr -\- grr -=. o ; multiply the Terms of that Scale refpedively by xXy x, i, (o as to have XX — frx -\- grr =r. o, let m and p be the two roots of that Equation, then having; made A = ■ ' ~ T and B z= ■■■ '""""' - , and ^ ' o m —p p — m ' fuppofing / to be the interval between the firfl Term and the place affigned, that Term will be Am' -\- Bp'. Secondly, If the Scale of Relation be fr — grr -^ hr^, make I — fr -\- grr — /6r3 = o, the Terms of which Equation being multiplied refpedtively by x\ xx, x, i, we fhall have the new E- quation x'^ — frxx -f- grrx — hr^ = 0, let m, p, q be the roots of that Equation, then having made A = '" — ,^ ^ '' ''" , p err — "'-\- '/ X ir -\- mqa _ err — /' -f" "» X ^r + mqa p — rn -x p — q ' q — m^q — p » And fuppofing as before / to be the Interval between the firfl: Term and the Term whofe place is afligned, that Term will be Km' -j- Bp' -1- Cq'. 'thirdly. If the Scale of Relation be fr — grr -j- hr^ — kr^ make i — fr -U grr — hr'^ ■\- kr^ = o, and multiply its Terms refpedively by x^, x\ xx, x, i, fo as to have the new Equation JC+ — frx^ -\- grrx'' — hr'x -f ^^* = o, let w, />, q, f, be roots of that Equation, then having made ^ dri —f^q-\.f X err +pq + p/+qf>i br — pq/ ■><■ a '" — p X ni — q x "1 ■ — f p tir'i — q + / + g; X err -\- qf-\- am -^ Jm v ^ r — pfm v <7 ■ q y p — / y p ■ — , Jri — / + 7/; 4- ;/ X err +Jm-\-/fj -\- mp X br — fmp x a q — / X q — "I •< Q — P — . dri — m -\- p-^ q V err -|- ?7ip - \- rr.g + fq \ hr — nipff y a f— m xj — p X. I — q then, flill fuppofing / to be the Interval between the firfl; Term and the Term whofe place is afligned, that Term will be hm' -j- Bp' -\- Altho' one may by a narrow infpedion perceive the Order of thofc Theorems, it will not be amifs to exprefs them in words at length. General Rule. Let the Roots ;;;, p, q, f, &cc. determined as above, be called re- G g fpedtively. 2 26 Tlje Doctrine <?/ Chances. fpedively, firft, fecond, third, fourth Root, &c. let there be taken as many Terms of the Series beginning from the firft, as there are parts in the Scale of Relation : then multiply in an inverted order, I °, the laft of thefe Terms by Unity j 2°, the laft but one by the Sum of the Roots wanting the firfl: ; 3", the laft but two, by the Sum of the Products of the Roots taken two and two, exclud- ing that produdl wherein the firft Root is concerned j 4°, the laft but three, by the Sum of the Produds of the Roots taken three and three, ftill excluding that Product in which the firft Root is concerned, and fo on ; then all the feveral parts which are thus ge- nerated by Multiplication being connefted together by Signs alter- nately politive and negative, will compofe the Numerator of that Fraction to which A is equal ; now the Numerator of that Fradlion to which B is equal will be formed in the fame manner, excluding the fecond Root inftead of the firft, and fo on As for the Denominators, they are formed in this manner: From the firft Root fubtradl feverally all the others, and let all the remain- ders be multiplied together, and the Produdl will conftltute the De- nominator of the Fradlion to which A is equal ; and in the fame manner, from the fecond Root fubtradting all the others, let all the remainders be multiplied together, and the Produdl will conftitute the Denominator of the Fradion to which B is equal, and fo on for the Reft. Corollary i. If the Series in which a Term is required to be afligned, be the Quotient of Unity divided by the differential Scale i — jr -^ grr — hr'^ -\- kr*, multiply the Terms of that Scale refpeftively by ^, x3, AT*, X, I, fo as to make the firft Index of x equal to the laft of r, then make the Produdl at* — frx^ -j- grrxx — hr^x -\- kr'* to be = o. Let as before w, p, q, J, be the Roots of that Equation, let alfo z be the number of thofe Roots, and / the Interval between the firft Term, and the Term required, then make Z — I q— m y. q — py.q — f j — m -f. f — p % f - and the Term required will be Aw' -^ B/>' ■\- Cy' -J- Tiq' ; and the Sun\ of the Terms will be Ax The Doctrine (?/'Chances. 227 Ax '-" -t-Bx -i::-^— + Cx '-^ +Dx-^^.- It is to be obferved, that the Interval between the firft Term and the Term required is always meafured by the number of Terms want- ing one, fo that having for Inftance the Terms, a, b, c, d, e, f, whereof a is the firft and / the Term required, the Interval be- tween a andy is ^, and the Number of all the Terms 6. Corollary 2. If in the recurring Series a -\- br -\- err ■\- dr^ -}- fr*, &c. where- of the Differential Scale is fuppofed to be i — fr\- grr — Zr' -\- /&r+, we make x^ — fxr^ -f- grrxx — hr^x -|- kr^ = o, and that the Roots of that Equation be m, p, q, f, and that it fo happen that fo many Terms of the Series a ■\- br -\- err -\- dr^ -f- er^, &c. as there are Roots, be every one of them equal to Unity, then any Term of the Series may be obtained thus ; let / be the Interval be- tween the firft Term and the Term required, make P X I — yxi — / T) I — y XI 7x1 A 1 — p X I — yxi — / T> ' — y X I m — p x. m — q \ m — f p — y x p — ,/ x p — m p, 1 — /x 1 — CT X I — p pj I — mx I — p X I — 7 q — y"x q — m y. q — /> ' / — m x / — p x / — q and the Term required will be Ajji^ -|- Bp' -\- Qq- -{- W- Proposition VII. If there be given a recurring Series whofe Scale of Relation is fr — grr, and out of that Series be compofed two other Series, whereof the firft fhall contain all the Terms of the Series given which are pofited in an odd place, and the fecond ftiall contain all the Terms that are pofited in even place ; then the Scale of Relation in each of thefe two new Series may be obtained as follows : Take the differential Scale i — Jr -\- grr, out of which compofe the Equation xx — frx -Y grr = o ; then making xx r= x, expunge the Quantity x, whereby the Equation will become 2 — fr\/ •z-\- grr = 0, or z -\- grr ^= fr \/ z ; and fquaring both parts, to take away the Radicality, we fliall have the new Equation zz -\- 7grrz ■\- ggr* = fj^f^y or zz -j- zgrrz \ ggr* = o ; and dividing its Terms refpedively by zz, z, i, we fliall have a new differential Scale for each of the two new Series into which the Series given was divided, which will be i -|- 2grr ■\- ggr* : and this being ob- — ffrr G g 2 tained, 228 'The Doctrine c*/' Chances. tained, it is plain from our firji Propojitlon, that each of the two new Series may be fummed up. But if the Scale of Relation be extended to three Terms, fuch as the Scale Jr — grr -{- hr^, then the differential Scale for each of the two Series into which the Series given may be fuppofed to be divided, will be i — Jfrr — 2fhr^ — hhr'', whereby it ap- J^ 2grr -\- ggr^ pears that each of the two new Series may be fummed up. If inftead of dividing the Series given into two Series, we di- vide it into three, whereof the firfl fliall be compofed of the i""', 4*'', 7'^', 10''', &;c. Terms; the fecond of the ^i^ 5''', 8''', 11''', &c. Terms; the third of the 3'', 6''', 9'^, la'"", &c. Terms; and that the Scale of Re- lation be fuppofed fr — grr ; then taking the differential Scale 1 — ff'-\- grr, and having out of it formed the Equation xx — Jrx 4" grr = o, fuppofe a.' =r 2 ; let now x be expunged, and the Equation will be changed into this zz -\- ifgr'^z ■\- gr^ z=. o, — pr'z of which the Terms being divided refpedively by zz^ z, i, we fliall have a differential Scale 1 — fr^ -\- gV^, which will ferve for every one of the three Series into which the Series given is di- vided ; and therefore every one of thofe three Series may be fummed up, by help of the two firfl Terms of each. If the Scale of Relation be compofed of never fo many parts, ftill if the Series given be to be divided into three other Series; from the fuppofition of -V being made = z, will be derived a Scale of Relation for the three parts into which the Series given is to be di- vided. But if the Series given was to be divided into 4, 5, 6, 7, &c. Series given, fuppofe accordingly x'^ = 2;, x^ =^ z, x^ = z, x'^ =z z, &:c. and x being expunged by the common Rules of Algebra, the Scale of Relation will be obtained for every one of the Series into which the Series given is to be divided. Proposition VIII. If there be given two Series, each having a particular Scale of • Relation, and that the correfponding Terms of both Series be added together, fo as to compofe a third Series, the differential Scale for this third Series will be obtained as follows. Let 'Th& Doctrine <^Chances. 229 Let I — fr -\- grr be the differential Scale of the firft, and 1 — mr -|- frr, the differential Scale of the fecond ; let thofe two Scales be multiplied together, and the Produd i — m •\- f y.r -\- p -y g -\~ "if ^ ^'^ — '''^ -\- pf Xr^ -\- pg X r*, will exprefs the differential Scale of the Series refulting from the addition of the other two. And the fame Rule will hold, if one Series be fubtraded from the other. Proposition IX. If there be given two recurring Series, and that the correfpond- ing Terms of thofe two Series be multiplied together, the differen- tial Scale of the Series refulting from the Multiplication of the other two mav be found as follows. Suppofe I — fr -|- grr to be the differential Scale of the firft, and I — ma Ar paa the differential Scale of the fecond, fo that the firft Series fhall proceed by the powers of r, and the fecond by the powers of a ; imagine thofe two differential Scales to be Equations equal to nothing, and both r and a to be indeterminate quantities ; make ar^=z, and now by means of the three Equations, i — fr -\- grr = 0, I — j}ja -\- paa =0, ar:= z, let both a and r be expunged, and the Equation refulting from that Operation will be I ^fmz ■\-ffpzz ^fgmpz' -f- ggppz^ = o -j- mmgzz — 2g[>ZZ or I — fmar -\- ffpa-r^ — fgmpa^r^ -}- ggppa'^r'^ =r o -|- mmga^r- — 2gpa'^r'^ by fubftituting ar in the room of z ; and the Terms of that Equa- tion, without any regard to their being made =: o, which was- purely a fidlion, will exprefs the differential Scale required : and in the fame manner may we proceed in all other more compound Cafes. But it is very obfervable, that if one of the differential Scales be the Binomial i — a raifed to any Power, it will be fufficient to raife the other differential Scale to that Power, only fubftituting ar for r, or leaving the Powers of r as they are, if a be reftrained to Unity ; and that Power of the other differential Scale will confti- tute the differential Scale required. Sotng 230 T^s Doctrine 0/ Chances. ^ome Ufes of the foregoing Propofitiom. We have feen in our lviii'*' Problem, that if two Adverfaries, whofe proportion of Skill be as a to b, play together till fuch time as either of them wins a certain number of Stakes, fuch as 4 for inftance, the Probability of the Play's not ending in any given num- ber of Games will be determined by -^. for 4 Games. ^ •'-T--,x6 lor 6 Games. « + "> ==N8 lOJ" o Games. — , „o for 10 Games. a + b \ -^ - -^-rv.J for 12 Games. a-\-b\ &C. Wherein it is evident that each Term in each of the three Columns written above is referred to the two preceding by a conftant Scale of Relation, fo that if the Terms of the firft Column which are iaH \ia*l>/> 4^a'ti ib^a^i* i;(oa-l>< r o- l^JV' ZW' ^TTV' M^TT- IT^^ ' ^^' b^ refpedlive- ly called E, F, G, H, K, &c. and that for fliortnefs fake we fuppofe = r, we (hall find G = ^rF — 2rrE, H = 4rG — 2rrF, ab and To on ; and therefore confidering the Sum of every three Terms whereby each Probability is exprefled as one fingle Term, and de- noting thofe Sums refpeftively by S, T, U, X, &c. we fhall find U = 4rT — 2rrS, X = ^rV — - 2rrT, and fo on ; from which it follows that the Method of determining the Probability of the Play's not ending in any number of Games given, is no more than the finding of a Term in a recurring Series. Let it therefore be required to find the Probability of 4 Stakes not being loft in 60 Games , to anfwer this, let it be imagined that the Probabilities of not ending in o, 2, 4, 6, 8, ID 60 Games, are exprefled by C, D, E, F, G, H, K refpedively ; then calling / the number of Games given, it is evident that the Term K is diftant from the Term C by an Interval r= — /, in this Cafe = 30, the odd numbers being omitted, by reafon it is impofli- ble !/:^£? Doctrine <j/ Chances. 231 ble an even number of Stakes fhould be won or loft exadly in an odd number of Games : moreover it being a certainty that the Set of 4 Stakes to be won or loft can neither be concluded before the Play begins, nor when no more than two Games are played off, it follows that the two Terms C, and D, are each of them equal to Unity ; for which reafon, if out of the Scale of Relation ^r — 2rr, or rather out of the differential Scale i — ^.r -\- 2rr, we form the Equation, xx — /^.rx -\- 2rr = o, and that the roots of that Equa- tion be f?i and p, and then make A =. ^^~^ , B = ^_ J ' , the two Terms alone Am ^ -]- Bp^ ' will determine the Probability required. This being conformable to Corollary 2^ of our vi''' Pro- portion, it will be proper to confult it. But becaufe in higher Cafes, that is when the number of Stakes to be won or loft is larger, it would fometimes be infinitely labori- ous to extradl the Roots of thofe Equations, it will be proper to fhew how thofe Roots are adually to be found in a Table of Sines. Of which to give one Inftance, let it be propofed to find the Pro- bability of the Play's not ending in any number of Games /, when the number of Stakes to be won or loft is 6 •, then arguing in the lame manner as in the preceding Cafe, let the Probabilities of the Play's not being concluded in o, 2, 4, 6, 8, 10 /Games be refpedively D, E, F, G, H, K z; then we may conclude that the three Terms D, E, F ftanding refpeftively over-againft the number of Games o, 2, 4, are each of them equal to Unity, it being a certainty that the Play cannot be concluded in that number of Games. Wherefore having taken the differential Scale I — 6r -\- grr — 2r'^, which belongs to that number of Stakes 6, and formed out of it the Equation at' — 6rxx -f- grfx — 2r^ z=z o,, let the Roots of that Equation be denoted by w, p, q ; then making A= jEZ^~ , B= ^ZL^"^^ , C== ^~^--~-P ^ the m — f >. m — q f — ?x^ — m q — m ^i. q — f -I ~l -I Probability required will be km'"' -|- B/)- + Cy- Now I fay that the Roots w, /», q of the Equation above written, may be derived from a Table of Sines ; for if the Semi-circumfe- rence of a Circle whofe Radius is 2r, be divided into 6 equal parts,. and we take the Co-verfed Sines of the Arcs that are -^ , t , — ' 6 ' o of the Semi-circumference, fo that the Numerators of thofe Frac- tions be all the odd numbers contained in 6, thofe Co-verfed Sines willi 232 I'be Doctrine <?/" Chances. will be the Values of w, f^ q, and the Rule is general and ex- tends to all Cafes ; Itill it is obfervable that when the number of Stakes is odd, for Inftancc 9, we ought to take only — , — , — , ■^ of the Semi-circumference, and rciedt the lafl: Term — ex- prefling the whole Semi-circumference. But what ought chiefly to recommend this Method is, that fup- pofing m to be the greateft Co-verfcd Sine, the firft Term alone -/ hm^ will give a fufficient approximation to the Probability re- quired, efpecially if / be a large number in itfelf, and it be alfo large in refpeft to the number of Stakes. Still thele Rules would not be eafily pradicable by reafon of the great number of Fadlors which might happen to be both in the Numerator and Denominator to which A is fuppofed equal, if I had not, from a thorough infpedtion into the nature of the Equations which determine the Values of ;«, p, q, Sec. deduced the following Theorems. 1', If n reprefents the number of Stakes to be won or loft, whether that number be even or odd, then the Numerator of the Fradlion to which A is equal, 'viz. 1 — px 1 — qx 1 — fx i — t, '"4- I" &c. will always be equal to the Fradion - - , — . > and in the fame manner that the Numerator of the Fradion to which B is equal, viz. I — y X I — /X I — t, &CC. will always be equal to the Fradion " ■ - — ==- , and fo on. 2°, If n be an even number, and that m be the right Sine correfponding to the Co-verfed Sine m; t hen the Denominator of the Fradion to which A is equal, viz. m — p x m — q X tn — fx m — /, —n &c. will always be equal to the Fradion —, ; and in the fame manner if p reprefent the right Sine belonging to the Co-verfed Sine />, then the De nomina tor of the Fradion to which B is equal, •viz. p — q y.p — J\p — t, &c. will always be equal to the Fradion I —71 -, — , and fo on. 3°, If 72 be an odd number, and that in be, as before, the right Sine correfponding to the Co-verfed Sine ni; then the Denomi- nator 'The Doctrine<?/"Chances. 233 -n nr- ill nator of the Fradion to which A is equal will be - ^^^^^^ > and the Denominator of the Fradlion to which B is equal will be I —n Corollary From all which it follows, that the Method of determining the Probability of a certain number n of Stakes not being loft in a given number / of Games, may be thus exprcffed. Let L be fuppofed = -"^^^^l , and r ==y- , then that Probability will be into X m — -^ — - ^p n I T — m I — p i — p ^ — -f &c. when n is an even number, or /— j_ /— I /-M '— i L . tn ^ tn 2 p' \J p 2 , q si q 2. s\ . "7" into x;« — ' — —y.p -\-- -y.q J.„ I — m i—p ^ ' I — q ^ \—s &c. when n is an odd nnmber. til 'ill 1/ 1/ But becaufe m '^ x s/m, p ^ x -Jp, &c. are the fame zs m^ ,p- refpedtively, it is plain that both Cafes are reduced to one and the fame Rule. It was upon this foundation that I prefcribed the Rule to be fecn in my lxix'*" Problem, wherein I did not diftinguifli the odd Cales from the even. But altho' the Rule there given feems fomewhat different from what it is here, yet at bottom there is no difference ; it confifting barely in this, that whereas 2r in this place is the Radius of the Circle to which the Calculation is adapted, there it is Unity, and that there the Co-verfed Sines were expreffed by their Equivalents in right Sines ; there was alfo this little difference, that the Denomi- nators 1 — -m, I — p, &c. were expreffed by means of the verfed Sines of thofe Arcs, to which m and p are co-verfed Sines. Other Variations might be introduced, fuch for inftance as might arife from the confideration of Vwr, -Jpr, &c. being the right H h Sines 2 34 T^& Doctrine t/ Chances. Sines of — the Complements to a Quadrant of the Arcs originally taken. But to fliew the farther ufe of thefe Series, it will be con- venient to propofe a Problem or two more relating to that Sub- je<a. PROBLEM LXX. M and N, nsohofe proportion of Chances to wi?i 07ie Game are refpe&ively as a to b, refolve to play together till one or the other has loji 4. Stakes : two Standers hyy R and S, co7icern themfelves in the Play^ R takes the Jide of M, and SofN, and agree betwixt thefn, that ^ JJjall fet to S, the Smn L to the Sum G on the frfl Game^ 2L to 2G on the fecond^ 3L /o 3G on the thirdy 4L to 4G on the fourth^ and in cafe the Play be ?iot then cojicltided^ 5L to 5G on the fifths and fo i?icrea- fng pe?'petually in /Arithmetic Progreffton the Sums which they are to fet to 07te aitother^ as loiig as M and^ play ; yet with this farther condition^ that the SumSy fet down by them R a?id S, pall at the e?td of each Game be taken up by the Winner^ afid not left upon the Tabh to he taken up at once upoji the Conclufton of the Play : it is demanded how the Gain of R is to be eflimated be- fore the Play begins. Solution. Let there be fuppofed a time wherein the number /> of Games has been played j then R having the number a of Chances to win the Sum p -Y 1 X G in the next Game ; and S having the number b of Chances to win the Sum /> -j- i x L, it is plain that the Gain of R in that circumftance ought to be eftimated by the quantity ^ -f- I X " '~ - ^ ; but this Gain being to be eftimated before the Play begins, it follows that it ought to be eftimated by the quantity p -\- lY. " ~^ — multiplied by the refpeilive Probability there l%e Doctrine (7/"Chances. 235 there is that the Play will not then be ended ; and therefore the whole Gain of R is the Sum of the Probabilities of the Play g not ending in o, i, 2, 3, 4, 5, 6, 6cc. Games in infinitum, multiplied by the refpedlive Values of the quantity /» -|- i x "^T^ , p being interpreted fucceffively by the Terms of the Arithmetic Progrefiion, o, I, 2, 3, 4, 5, 6, &c. Now, let thefe Probabilities of the Play's not ending be refpedively reprefented by A, B, C, D, E, F, G, I, &c. let alfo the Quantity - J~^ '- be called S, and then it will follow that the Gain of R will be expreffed by the Series AS -\- 2BS -I- 3CS -1- 4DS 4-5ES + 6FS + 7GS, &c. but in this Problem, altho' the Probabilities of the Play's not ending decreafe continually, yet the number of Stakes being even, the Probability of the Play's not ending in an odd number of Games is not lefs than the Probability of not ending in the even number that imme- diately precedes the odd j and therefore B:=zA, D = C, F — E I = G, &c. from whence it follows that the Gain of R will be ex- preffed by the produd: of S /';//(? 3A-|-7C-f-iiE4-i 5G -|- 1 9I, &c. but the differential Scale for the Series A -j- C -|- E -j- G, &c. is I — 4r H- zrr, wherein r is fuppofed = - ^T-^, , and the diffe- rential Scale for the Series 3-|-74" ^^ ~l" '5"l" I9> &c. is i — i 3^-1-3^^ — fl', wherein ^ = 1. And therefore the differential Scale for the Series 3A-4-7C-1-- uE, &c. confifting of the pro- duds of the Terms of one Series by the correfponding Terms of the other, will be i — 4^-1- 2?7A% or i — 8r -j- 20;r — i6r' -\- ^r*; and therefore having written down the four firfl Terms of the Series to be fummed up, viz. as many Terms wanting one as there are in the differential Scale, multiply them in order by the differen- tial Scale according to the prefcription given in the Remark be- longing to our third Propofition, and the Produdl will be the Nu- merator of the Fraftion expreffing the Sum, of which Fradion the Denominator will be i — 4;- -\- zrr^ *; But to make this the plainer, here follows the Operation, 3A-f 7C-f iiE-f 15G I — 8V -\- zorr — i6r' 3AH- 7C-I- iiE -\- 15G — 24rA— 56rC— 88rE . ■\-6orrh-\- i^orrC . . — 48r'A. II h 2 And 236 T^e Doctrine 0/" Chances. And thus is the Numerator obtained : but A = j , it being a cer- tainty that the Play cannot be ended before it is begun, and C is liicewife = i, it being a certainty that 4 Stakes cannot be loft neither before nor at the expiration of 2 Ganaes ; but by the law of Relation of the Terms of the Series, E = \rQ, — 2rrA, and G =:: 4rE — 2rrC, and therefore the proper Subflitutions being made, 10 — -,6r-|- :;Orr + 8r3 the Sum of the Series will be found to be S into — x— , + zrr\- and now in the room of S and r fubftituting their refpedtive Values -H^rr-^"'^ -TTTT tl^e Sum , + ^ tf^to will exprefs the Gain of R. Corollary i. If the Stake L be greater than the Stake G, in the fame pro- portion as a is greater than 4 there can be no advantage on either fide. Corollary 2. If rt and 6 arc equal, the Gain of 7? will be 216 times the half difference between the Stakes G and L : thus if G ftands for a Guinea of 21'* and L for 20/- the Gain of R will be 216 Six- pences, that is, 5^ — 8/^- Corollary 3. If a be greater than I^, the Gain of R, according to that inequa- lity, will vary an infinite number of ways, yet not be greateft when the proportion of d; to ^ is greateft ; fo that for Inftance, if the proportion of a to /? is 2 to i, and G and L are equal, the Gain of R will be about 29 -G; but if « is to /^ as 3 to i, the Gain of R will be no more than about 22 — G ; and if the proportion of a to b be infinitely great, which would make R win infallibly, the Gain of R will be only 10 G. But altho' this may feem at firft a very ftrange Paradox, yet the reafon of it will eafily be apprehended from this confideration, that the greater the proportion is of a to ^, fo much the fooner is the Play likely to be concluded ; and therefore if that proportion were infinite, the Play would neceffarily be ter- minated in 4 Games, which would make the Gain of 2?! to be i -}- But The Doctrine o/"Chances. 237 But if It was required what muft be the proportion of ^ to 3 which will afford to K the greateft advantage poffible, the anfwer will be very near 2 to i, as maybe found eafily upon Trial; and maybe found accurately by the Method which the Geometricians call de Maximis & Minims. PROBLEM LXXI. If M and N, w/joje number of Chances to win one Game are refpeSiively as a io b, p/aj together till four Stakes are won or loji on either fide ; afid that at the fame timey R and S whofe ?mmber of Cha?ices to win one Game are refpeSiively as c to d, play alfo together till fve Stakes are won or lofl on either fide ; what is the Probability that the Play between M and N will be ended in fewer Games ^ than the Play between R and S. Solution. The Probability of the firft Play's being ended in any number of Games before the fecond, is compounded of the Probability of the firft Play's being ended in that number of Games, and of the fe- cond's not being ended with the Game immediately preceding : from whence it follows, that the Probability of the firft Play's end- ing in an indeterminate number of Games before the fecond, is the Sum of all the Probabilities in infinitum of the firft Play's ending, multiplied by the refpedlive Probabilities of the fecond's not beino- ended with the Game immediately preceding. Let A, B, C, D, E, &c. reprefent the Probabilities of the firft Play's ending in 4,6, 8, 10, 12, &c. Games rcfpedively ; let alfo F, G, fl, K, L, &c. reprefent the Probabilities of the fecond's not being cndcc' in 3, 5, 7, 9. 11, &c. Games refpecftively : hence, by what we have laid down before, the Probability of the firft Play's end- ing before the fecond will be reprefented by the infinite Series AF -1- BG + CH + DK + EL, &c. Now to find the Law of Rela- tion in this third Series, we muft fix the Law of Relation in the firft and fecond, which will be done by our tx"" Problem, it being for the firft \r — 2rr, wherein r is fuppofed = ^ — -^r " ; and becaufe, as we have obferved before, the Law of Relation in thofe Series. 238 lie Doctrine <?/" Chances. Scries which exprefs the Probability of not ending, is the fame as the Law of Relation in the refpedlive Series which exprefs the Pro- bability of ending; it will alfo be found by the direftions given in our Lx'*" Problem, that if we fuppofe -^~rr = w, the Law of Relation for the fecond Series will be ^m — 5;;;^, and therefore the Laws of Relation in the firll: and fecond Series will refpedively be I — 4r -\- zrr, i — 5//; -\- ^mm. And now having fuppofed thofe two differential Scales as Equations = o, and fuppofed alfo rm^=.z, we fl:all find by the Rules delivered in our ix''' Propo- fition, that the Scale of Relation for the third Series will be i — 20Z ■\- I lozz — 20o,'23 -\- looz'^ ; and therefore having taken the four firft Terms of the third Series, and multiplied them by the differential Scale, according to the proper Limitations prefcribed in our 1 1 1'' Propofition, we fliall find the Sum of the third Series to be AF + BG +CH +DK — 20AFG2;— 20BG2; — 20CH2; -|- 1 1 o AFsz -{- 1 1 cBGzz — aooAFz^ 1 2OZ -j- I I ox* — 200Z5 _j_ iQQz'^ a4 -I- (,* Now fuppofing S to reprefent the Fradiion ' — -t-^v , the four Terms A, B, C, D will be found to be iS -\- 4^8 -j- i4rrS + 481-3 S ; but the four Terms F, G, H, K wherein S is not concerned will be found to be i, 5/;/ — Sww, 20ww — ^^tn'^, J S'"^ — loow''-; and therefore the proper Subftitutions being made in the Sum above written, we (hall have that Sum reduced to its proper Data ; and that ^um thus reduced will exhibit the Probability required. But becaufe thofe Data are many, it cannot be expedted that the So- lution fliould have fo great a degree of Simplicity as if we had re- flrairied a and ^ to a ratio of Equality, which if we had, the Pro- bability reqpired would have been exprefTed by the Fradion ,z — iozz ,~ ___ 1^^^ becaufe r has been fuppofed 1 — lOZ + I lO^S-. 20Ca> -|- 100~ L =:r - -L- - — , it follows that r in this Cafe is ^=: — : and again, a-\-b\ 4 becaufe ;;; has been fuppofed ==: ■— ^r , then m is alfo z= — , for which reafon rm or z-=z --^ , for which reafon fubflituting — !- inftcad of z, the Probability required will be exprefTed by the Fraction 1*he Doctrine of Chances. 239 Fracftlon -i^ : Now fubtraftins; this Fradion from Unity, the remainder will be the Fradion -^^ , and therefore the Odds of the firfl: Play's ending before the fecond will be 476 to 247, or 27 to 14 nearly. PROBLEM LXXII. A and B playing together^ and having an equal number of Chances to win one Game^ engage to a SpeBator S that after an even number of Games n is over^ the Wi7iner fhall give him as many Pieces as he wins Games over and above one half the number of Games playedy it is demaftded how the Expe&atio?i of S is to be determined. Solution. Let E denote the middle Term of the Binomial a-\-b raifed to the Power 7/, then l — will exprefs the number of Pieces 2" which the Spedator has a right to exped, Thus fuppofing that A and B were to play 6 GameSj then rai- ling a ■\' b \.o the 6'*' Power, all the following Terms will be found in it, 'u/z. a^ -f- ba'>b -\- 1 S^'^bh -\-2oa^b' -\- 1 ^aab'^ -j- 6ab^ + b''. But becaufe the Chances which yi and B have to win one Game have been fuppofed equal, then a and b may both be nude =r i, which will make it that the middle Term E will be 20 ; there- fore this number being multiplied by -«, that is in this Cafe by 3, the Produd will be 60, which being divided by 2- or 2', that is by 64, the Quotient will be -^ or — , and therefore the Expedation of S is as good to him as if he had ~ of a Piece given him, and for that Sum he might transfer his Right to an- other. It will be eafy by Trial to be fatisfied of the Truth of this Con- clufion, for refuming the 6''' Power of a -\- b, and confidering the firft Term a^, which Oiews the number of Chances for A to win 6 times ; in which Cafe S would have 3 Pieces given him, then the Expeda. 240 "The Doctrine (t/" Chances. Expedation of S arifing from that profpedt is ^^^^—^^f > ^^^^ '^^ T" ' confidering next the Term ba^I^ which denotes the number of Chances for A to win 5 times and lofing once, whereby he would get two Games above 3, and confequently S get 2 Pieces, then the Expedation of 5 arifing from that profpe(fl would be '}rrpi^' or — ; laftly confidering the third Term 1 5 a*/>^ which fhews the nam- ber of Chances for A to get 4 Games out of 6, and confequently for S to get I Piece, the Expedation of S arifing from that pro- fpedl would be -^^y" or -^ , the fourth Term aort^i^J would afford nothing to 5, it denoting the number of Chances for A to win no more than 3 Games ; and therefore that part of the Ex- pedation of S, which is founded on the Engagement of A to him, would be — — '-^ ^^"oT ' ^"^ ^^ expeds as much from B, and therefore his whole Expedation [s ~ z= -~ as had been before determined. And in the fame manner, if A and B were to play 12 Games the Expedation of S would be -^^ , which indeed is greater than in the preceding Cafe, but lefs than in the proportion of the num- ber of Games played, his Expedation in this Cafe being to the for- mer as 5544 to ^840, which is very little more than in the propor- tion of 3 to 2, but very far from the proportion of 12 to 6, or 2 to 1. And if we fuppofe flill a greater number of Games to be played between A and B, the Expedation of S would flill increafe, but in a lefs proportion than before ; for inftance, if A and B were to play 100 Games, the Expedation of S would be 3.9795 ; if 200, 5.6338 ; if 300, 6.9041 ; if 400, 7.9738; if 5C0, 8.9161 ; if 700, 800, 900, IO.CC2, 11.280, 11.965 refpedively, fo that in 100 Games the Ex- pedation of 5 would be in refped to that number of Games about — , and in 900 Games that Expedation would not be above -:rr- • Now how to find the middle Terms of thofe high Powers will be fhewn afterwards. Corollary, From the foregoing confiderations, it follows, that if after taking a great number of Experiments, it lliould be obferved that the hap- penings Ihe Doctrine oJ Chances. 241 penlngs or failings of an Event have been very near a ratio of Equa- lity, it may fafely be concluded, that the Probabilities of its happen- ing or failing at any one time affigned are very near equal. PROBLEM LXXIII. A and B playing together^ and having a different num- ber of Chances to win one Game^ which number of Chances I fuppofe to be refpetlively as a to b, engage themf elves to a Spe&ator S, that after a certain nmn- ber of Games is over, A fhall give hitn as many Pieces as he wins Ga?nes. over and above — v-T-n^ cind B as ' a-\-b ' many as he wins Gaines, over and above the numher - , ; 71 ; to find the Expe&ation of S. Solution. Let E be that Term of the Binomial iiArb raifed to the Power w, in which the Indices of the Powers of a and b (hall be in the lame ratio to one another as « is to /^ ; let alfo p and q denote refpedlively thofe Indices, then will the Expedtation of S from A and B together be ■ — ==r-'E, or — — !— ^„ E from either of them in particular. Thus fuppofing the number of Games n to be 6, and that the ratio of rt to ^ is as 2 to i ; then that Term E of the Binomial a -\- b raifed to its 6'*^ Power, wherein the Indices have the fame ratio to one another as 2 to i, is i i:,a'^b-, and therefore ^ = 4, and 5- = 2 ; and becaufe, ^, b, p, q, n are refpedtively 2, i, 4, 2, 6, thence the Expeftation — ^f-^- xE will be in this particular Cafe X240, n^a-\-t,\ ^ 4374 640 o , or = — nearly. But fuppofing that A and B refolve to play 1 2 Games, then that Term of the Binomial a ■\- b raifed to its 1 z'*" Power, wherein the Indices/" and ^ have the fame ratio as 2 to i, is 495^'*^*; and be- caufe the Quantities a, b, p, q, n, are refpeflively 2, i, 8, 4, 12, the Expedlation of S will be — d^iZ. or -^ nearly. And again, if A and B play ftill a greater number of Games, the Expedfation of S will perpetually increafe, but in a lefs proportion than of the number of Games played. I i COROL- >42 7^^ Doctrine (p/* C HA NCES. Corollary. From this it follows, that if after taking a great number of Expe- riments, it fhould be perceived that the happenings and failing"^ have been nearly in a certain proportion, fuch as of 2 to i, it may ijfely be concluded that the Probabilities of happening or failing at .iny one time afligned will be very near in that proportion, and that the greater tlie number of Experiments has been, fo much nearer the Truth will the conjedures be that are derived from them. But fuppofe it fliould be faid, that notwithftanding the reafonable- nefs of building Conjedures upon Obfervations, flill confidering the great Power of Chance, Events might at long run fall out in a dif- ferent proportion from the real Bent which they have to happen one way or the other ; and that fuppofing for Inflance that an Event might as eafily happen as not happen, whether after three thoufand Experiments it may not be poffible it (liould have happened two thou- fand times and failed a thoufand ; and that therefore the Odds againft fo great a variation from Equality fhould be affigned, whereby the Mind would be the better difpofed in the Conclufions derived from the Experiments. In anfwer to this, I'll take the liberty to fay, that this is the hardeft Problem that can be propofed on the Subjedl of Chance, for which reafon I have referved it for the lafl, but I hope to be forgiven if my Solution is not fitted to the capacity of all Readers ; however I fhall derive from it fome Conclufions that may be of ufe to every body : in order thereto, I fliall here tranflate a Paper of mine which was printed November 12, 1733, and communicated to fome Friend?, but never yet made public, referving to myfelf the right of enlarging my own Thoughts, as occafion fhall require. Novemb. 12, 1733. A Me- 'The Doctrine f?/" Chances. 243 A Method of approximating the Sum of the Terms of the Binomial 2. -\-h\^' expanded into a Series, from whence are deduced fome praclical Rules to eJUmate the Degree of Ajfent which is to be given to Experiments. ALT HO' the Solution of Problems of Chanc e often requires that feveral Terms of the Binomial a-\-b\ " be added to- gether, neverthelefs in very high Powers the thing appears fo laborious, and of fo great difficulty, that few people have un- dertaken that Tafk j for befides James and Nicolas Bernoulli, two great Mathematicians, I know of no body that has attempted it ; in which, tho' they have fhewn very great ikill, and have the praife which is due to their Induftry, yet fome things were farther re- quired ; for what they have done is not fo much an Approximation as the determining very wide limits, within which they demonftrated that the Sum of the Terms was contained. Now the Method which they have followed has been briefly defcribed in my Mifcellanea Ana- lytical which the Reader may confult if he pleafes, unlefs they ra- ther chufe, which perhaps would be the beft, to confult what they themfelves have writ upon that fubje^l: for my part, what made me apply myfelf to that Inquiry was not out of opinion that I fhould excel others, in which however I might have been forgiven ; but what I did was in compliance to the defire of a very worthy Gentleman, and good Mathematician, who encouraged me to it : I now add fome new thoughts to the former ; but in order to make their connexion the clearer, it is neceffary for me to refume fome few things that have been delivered by me a pretty while ago. I. It is now a dozen years or more fince I had found what fol- lows ; If the Binomial i -j- i be raifed to a very high Power de- noted by «, the ratio which the middle Term has to the Sum of all the Terms, that is, to 2', may be exprefled by the Fradion —, 7= , wherein A reprefents the number of which the Hy- » X t/ n— if •' perbolic Logarithm is -^ ^ — A ! ^-r— , &c. But be- » o i^ jco 1 1200 it8o ' I i 2 caufc 244 T'^^ Doctrine <?/' Chances. caufe the Quantity -^^ — \ — or i !-' is very nearly given when « is a high Power, which is not difficult to prove, it follows that, in an infinite Power, that Quantity will be abfolutely given, and reprelent the number of which the Hyperbolic Logarithm is — 1 ; from whence it follows, that if B denotes the Number of which the Hyperbolic Logarithm is — i -I ! ^l \ — •' * '-' ' \Z 3OO ' 1 2()0 &c. the Expreffion above-written will become — ~ — • and that therefore if we change the Signs of that Series, and now fuppofe that B reprefents the Number of which the Hyperbolic Logarithm is i —^-y-^^ J__|__l_ , &c. that Expreffion will be changed into -ytT "When I firft began that inquiry, I contented myfelf to determine at large the Value of B, which was done by the addition of fome Terms of the above-written Series ; but as I perceived that it con- verged but flowly, and feeing at the fame time that what I had done anfwered my purpofe tolerably vvdl, I defiftcd from proceeding far- ther till my worthy and learned Friend Mr. James Stirling, who had applied himfelf after me to that inquiry, found that the Quan- tity B did denote the Square-root of the Circumference of a Circle whole Radius is Unity, fo that if that Circumference be called c, the Ratio of the middle Term to the Sum of all the Terms will be exprefled by ^^- . But altho' it be not neceffary to know what relation the number B may have to the Circumference of the Circle, provided its value be attained, either by purfuing the Logarithmic Series before men- tioned, or any other way j yet I own with pleafure that this dif- covery, befides that it has faved trouble, has fpread a lingular Ele- gancy on the Solution. II. I alfo found that the Logarithm of the Ratio which the middle Term of a high Power has to any Term diftant from it by an Interval denoted by /, would be denoted by a ve ry near appr oxima- tion, (fuppofing m ■=^ -n) by the Quantities m -\- I — ~ X Log- „; _|- / _ 1 _|- ;?, — / -[- 1 X Log. m — l-\-\ — 2w X Log. m 4- Log. -^ C O R L-= Tide Doctrine ^Chances. 245 Corollary i. This being admitted, I conclude, that if m or ~n be a Quantity infinitely great, then the Logarithm of the Ratio, which a Term diftant from the middle by the Interval /, has to the middle Term, is 2// Corollary 2. The Number, which anfwers to the Hyperbolic Logarithm zll . . —, being ^ w ' znn On> "T~ 2inV xzon^ "• 720^* ' ' it follows, that the Sum of the Terms intercepted between the Middle, and that whofe diftance from it is denoted by /, will be — ; — into / ; T r- A , &C. »/nc ixin • 2-x.^nn Oxyai ' 24x9;)+ izoxlinS' Let now / be fuppofed =s^ny then the faid Sum will be ex- prcfled by the Series —7— into / — ■—- -A — ■ -r —^-^ — , &c. \/c J 3 ' 2x5 0x7 ' 24x9 1:0x11 ' Moreover, if /be interpreted by — , then the Series will become into L_.| 1 — \ — I : \ 8,- »/c 2 3x4 ' 2x5x8 6x7x10 I 24x9x32 120x11x64' which converges fo faft, that by help of no more than feven or eight Terms, the Sum required may be carried to fix or feven places of Decimals: Now that Sum will be found to be 0.427812, inde- pendently from the common Multiplicator -^ , and therefore to the Tabular Logarithm of 0.427812, which is 9.6312529, adding the Logarithm of -7^, v;z. 9.9019400, the Sum will be 19.533 1929, to which anfwers the number 0.341344. L E M MA. If an Event be fo dependent on Chance, as that the Probabilities of its happening or failing be equal, and that a certain given number // of Experiments be taken to obferve how often it happens and fails, and alfo that / be another given number, lefs than -j/;, then the Pro- bability of its neither happening more frequently than ~n -|- / times,. 246 lloe Doctrine o/" Chances. times, nor more rarely than — « — / times, may be found as fol- lows. Let L and L he two Terms equally diftant on both fides of the middle Term of the Binomial i -\- A " expanded, by an Inter- val equal to /; let alfo /" be the Sum of the Terms included between L and L together with the Extreams, then the Probability required will be rightly expreffed by the Fradlion -^ j which being founded on the common Principles of the Dodtrine of Chances, requires no Demonflration in this place. Cor o l l ary 3. And therefore, if it was poflible to take an infinite number of Experiments, the Probability that an Event which has an equal number of Chances to happen or fail, fhall neither appear more frequently than -n A^ —-J n times, nor more rarely than —n — — n/« times, will be expreffed by the double Sum of the number exhibited in the fecond Corollary, that is, by 0.682688, and con- fequently the Probability of the contrary, which is that of hap- pening more frequently or more rarely than in the proportion above affigned will be 0.3 173 12, thofe two Probabilities together com- pleating Unity, which is the meafure of Certainty: Now the Ratio of thofe Probabilities is in fmall Terms 28 to 13 very near. Corollary 4. But altho' the taking an infinite number of Experiments be not pradicable, yet the preceding Conclufions may very well be applied to finite numbers, provided they be great : for Inftance, if 3600 Ex- periments be taken, make « = 36oo, hence -n will be = 1800, and -V'^^ 30, then the Probability of the Event's neither appearing oftner than 1830 times, nor more rarely than 1770, will be 0.682688. COROLL ARY 5. And therefore we may lay this down for a fundamental Maxim, that in high Powers, the Ratio, which the Sum of the Terms in- cluded between two Extreams diftant on both fides from the middle Term by an Interval equal to - •Jn^ bears to the Sum of all the T%e Doctrine o/" Chances. 247 the Terms, will be rightly expreffed by the Decimal o 682688, that is — nearly. 4; , / Still, it is not to be imagined that there is any neceflity that the number n fhould be immenfely great ; for fuppofing it not to reach beyond the 900''' Power, nay not even beyondkthe 100''', the Rule here given will be tolerably accurate, which I have had confirmed by Trials. But it is worth while to obferve, that fuch a fmall part as is —\ln in rcfpedt to «, and fo much the lefs in refpedl to n as n increafes, does very foon give the Probability -^ or the Odds of 28 to 13 ; from whence we may naturally be led to enquire, what are the Bounds within which the proportion of Equality is contained ? I anfwer, that thefe Bounds will be fet at fuch a diftance from the middle Term, as will be expreffed by - V 2« very near ; fo in the Cafe above mentioned, wherein n was fuppofed =3600, —v'aw will be about 21.2 nearly, which in refpeft to 3600, is not above — ^th part : fo that it is an equal Chance nearly, or rather fome- thing more, that in 3600 Experiments, in each of which an Event may as well happen as fail, the Excefs of the happenings or failings above 1800 times will be no more than about 21. Corollary 6. If / be interpreted by k' n, the Series will not converge fo faft as it did in the former Cafe when / was interpreted by — \/«, for here no lefs than 12 or 13 Terms of the Series will afford a to- lerable approximation, and it would flill require more Terms, ac- cording as / bears a greater proportion to -J n\ for which reafon I make ufe in this Cafe of the Artifice of Mechanic Quadratures, firll: invented by Sir Ijaac Neuion, and fince profecuted by Mr. Cotes, Mr. James Stirling, myltlf, and perhaps others •, it confifls in de- termining the Area of a Curve nearly, from knowing a certain num- ber of its Ordinates A, B, C, D, E, F, &;c. placed at equal Intervals, the more Ordinates there are, the more exad will the Quadrature be ; but here I confine myfelf to four, as being fufficient for my purpofe : let us therefore fuppofe that the four Ordinates are A, B, C, D, and that the Diflance between the firfl and laft is denoted by /. then 24^^ Tlje Doctrine •t/'Chances. A t hen th e Ar ea con tained between the firfl and the lafl: will be — — ^ — — X /; now let us take the Diftances o\/», -^ n/ ;/, -^ v' ;/, -^-J n, \'J >i, -^ v/ ;;, ~ V ;/, of which every one exceeds the preceding by-A/«, and of which the lafl; is V?;; of thefe let us take the four lafl;, viz. —"J n, ^ "/ », •r^/«, — Vw, then taking their Squares, doubling feach of them, dividing them all by ;?, and prefixing to them all the Sign — , we fliall have '-, — - , — , — — , which mufl; be looked upon as Hy- perbolic Logarithms, of which confequently the correfponding num- bers, -u/z. 0.60653, 0.41 1 1 1, 0.24935, 0.13534 will ftand for the four Ordinates A, B, C, D. Now having interpreted / by — v/ ;i, the Area will be found to be t= 0.170203 xV», the double of which being multiplied by — 7;^- , the produdt will be 0.27160 ; let there- fore this be added to the Area found before, that is, to 0.682688, and the Sum 0.9,428 will ftiew what, after a number of Trials denoted by ;;, the Probability will be of the Event's neither hap- pening oftner than -77 4- V« times, nor more rarely than —« — \/«, and therefore the Probability of the contrary will be 0.04572 : which fhews that the Odds of the Event's neither happening oftner nor more rarely than within the Limits afligned are 2 i to i nearly. And by the fame way of reafoning, it will be found that the Pro- bability of the Event's neither appearing oftner than jfi -}- -j \/«, nor more rarely than —n — -v'?^ will be 0.99874, which will make it that the Odds in this Cafe will be 369 to i nearly. To apply this to particular Examples, it will be necefliiry to eftimate the frequency of an Event's happening or failing by the Square-root of the number which denotes how many Experiments have been, or are defigned to be taken ; and this Square-root, ac- cording as it has been already hinted at in the fourth Corollary, will be as it were the Modulus by which we are to regulate our Eftimation ; and therefore fuppofe the number of Experiments to be taken is •^600, and that it were required to affign the Probability of the Event's neither happening oftner than 2850 times, nor more rarely than 1750, which two numbers may be varied at pleafure, provided they be equally diftant from the middle Sum 1800, then make the half difference The Doctrine oJ Chances. 249 difference between the two numbers 1850 and 1750, that is, in this Cafe, 5o=/V«; now having fuppofed 3600 = «, tlien \/« will be ^ 60, which will make it that 50 will be = 60/, and confe- quently / = -^ = -^ j and therefore if we take the proportion, which in an infinite power, the double Sum of the Terms cor- refponding to the Interval \'J n, bears to the Sum of all the Terms, we fhall have the Probability required exceeding near. Lemma 2 . In any Power a-\- 0^" expanded, the greateft Term is that in which the Indices of the Powers of a and b^ have the fame propor- tion to one another as the Quantities themfelves a and b ; thus tak- ing the 10''' Power of a ■\- b^ which is a}° -\- loa'^b -\- \^a^b'^ -j- izoa'b'i -{• 2iort^M -\- 252^5^5 _|. zioa'^b^ -]- lao^'^^ 4- \S^'^b^ •\- loab"^ A^ b^° ; and fuppofing that the proportion of a to /^ is as 3 to 2, then the Term ^\oa'b'^ will be the greateft, by reafon that the Indices of the Powers of a and b^ which are iia that Term, are in the proportion of 3 to 2 ; but fuppofing the proportion of ^ x.q b had been as 4 to i , then the Term ^^^a^b"- had been the greateft. Lemma 3. If an Event fo depends on Chance, as that the Probabilities of its happening or failing be in any afligned proportion, fuch as may be fuppofed of a to ^, and a certain number of Experiments be de- ligned to be taken, in order to obfcrve how often the Event will hap- pen or fail ; then the Probability that it fliall neither happen more frequently than fo many times as are denoted by ^ -{- /, nor more rarely than fo many times as are denoted by —^, /, will be found as follows : Let L and R be equally diftant by the Interval / from the greateft Term ; let alfo S be the Sum of the Terms included between L and R, together with thofe Extreams, then the Probability required will be rightly exprefiJed by ■> ^ . Corollary 8. The Ratio which, in an infinite Power denoted by w, the greateft Term bears to the Sum of all the reft, will be rightly expreffed by K k the 250 Ithe Doctrine 6/" Chances. a->rl> tj abm of a Circle for a Radius equal to Unity. the Fradtion - "*"/ - , wherein c denotes, as before, the Circumference Corollary 9. If, in an infinite Power, any Term be diftant from the Greatefl: by the Interval /, then the Hyperbolic Logarithm of the Ratio which that Term bears to the Greateft will be exprefled by the Fraction — "'^abn ^^^' P'^ovi*^^^ ^he Ratio of / to « be not a finite Ratio, but fuch a one as may be conceived between any given number p and V «, fo that / be expreflible by /> V », in which Cafe the two Terms L and R will be equal. Corollary 10. If the Probabilities of happening and failing be in any given Ratio of inequality, the Problems relating to the Sum of the Terms of the Binomial a '\- i " will be folved with the fame facility as thofe in which the Probabilities of happening and failing are in a Ratio of Equality. Remark I. From what has been faid, it follows, that Chance very little di- fturbs the Events which in their natural Inftitution were defigned to happen or fail, according to fome determinate Law ; for if in order to help our conception, we imagine a round piece of Metal, with two polifhed oppofite faces, differing in nothing but their colour, whereof one may be fuppofed to be white, and the other black ; it is plain that we may fay, that this piece may with equal facility exhibit a white or black face, and we may even fuppofe that it was framed with that particular view of fhewing fometimes one face, fome- times the other, and that confequently if it be toffed up Chance fhall decide the appearance ; But we have feen in our lxxii'' Problem, that altho' Chance may produce an inequality of appearance, and flill a greater inequality according to the length of time in which it may exert itfelf, yet the appearances, either one way or the other, will perpetually tend to a proportion of Equality : But befides, we have feen in the prefent Problem, that in a great number of Experi- ments, fuch as 3600, it would be the Odds of above 2 to i, that cne of the Faces, fuppofe the white, (hall not appear more frequently than 1830 times, nor more rarely than 1770, or in other Terms, that The Doctrine ^Chances. 251 that it fliall not be above or under the pcrfedl Equality by more than ■7^ part of the whole number of appearances; and by the fame Rule, that if the number of Trials had been 14400 inftead of 3600, then ftill it would be above the Odds of 2 to j, that the appearances either one way or other would not deviate from perfedt Equality by more than -^ part of the whole : and in looooco Trials it would be the Odds of above 2 to i, that the deviation from perfedl Equality would not be more than by -~^ part of the whole. But the Odds would increafe at a prodigious rate, if inftead of taking fuch narrow limits on both fides the Term of Equality, as are re- prefented by ^ V », we double thofe Limits or triple them ; for in the firft Cafe the Odds would become 21 to i, and in the fecond 369 to I, and ftill be vaftly greater if we were to quadruple them, and at laft be infinitely great ; and yet whether we double, triple or quadruple them, ^c. the Extenfion of thofe Limits will bear but an inconfiderable proportion to the whole, and none at all, if the whole be infinite ; of which the reafon will eafily be perceived by Mathema- ticians, who know, that the Square-root of any Power bears fo much a lefs proportion to that Power, as the Index of it is great. What we have faid is alfo applicable to a Ratio of Inequality, as appears from our 9"^ CoroUar}'. And thus in all Cafes it will be found, that.tz/Z/'o' Chance produces Irregularities, ftill the Odds will be infinitely great, that in procefs of Time, thofe Irregularities will bear no proportion to the recurrency of that Order which naturally refultsfrom Original Design. Remark II. As, upon the Suppofition of a certain determinate Law according to which any Event is to happen, we demonftrate that the Ratio of Happenings will continually approach to that Law, as the Expe- riments or Obfervations are multiplied : fo, converfely, if from num- berlefs Obfervations we find the Ratio of the Events to converc^e to a determinate quantity, as to the Ratio of P to Q; then we conclude that this Ratio expreflTes the determinate Law according to which the Event is to happen. For let that Law be exprefi*ed not by the Ratio P : Q, but by fome other, as R : S ; then would the Ratio of the Events converge to this laft, not to the former : which contradids our Hypothefis. And the like, or greater, Abfurdity follows, if we fliould fuppofe the K k 2 Event 252 T^e Doctrine tf/"CHANCEs. Event not to happen according to any Law, but in a manner altoge- ther defultory and uncertain ; for then the Events would converge to no fixt Ratio at all. A^ain, as it is thus demonftrable that there are, in the conftitu- tion of things, certain Laws according to which Events happen , it is no lefs evident from Obfervation, that thofe Laws ferve to wife, ufe- ful and beneficent purpofes ; to preferve the ftedfaft Order of the Univerfe, to propagate the feveral Species ot Beings, and furnifh to the fentient Kind fuch degrees of happinefs as are fuited to their State. But fuch Laws, as well as the original Dcfign and Purpofe of their Eftablifliment, muft all be frovi without -, the Inertia of mat- ter, and the nature of all created Beings, rendering it impoffible that any thing Ihould modify its own effence, or give to itfelf, or to any thing elfe, an original determination or propenfity. And hence, if we blind not ourfelves with metaphyfical duft, we fliall be led, by a {hort and obvious way, to the acknowledgment of the great Maker and Governour of all j Him/elf all-iaife, all-powerful and good. Mr. Nicolas Bernoulli *, a very learned and good Man, by not connefting the latter part of our reafoning with the firft, was led to difcard and even to vilify this Argument from Jinal Caufes, fo much infifted on by our beft Writers ; particularly in the Inflance of the nearly equal numbers of male and female Births, adduced by that excellent Perfon the late Dr. Arbuthnot, in Vhil Tranf N°. 328. Mr. Binicul/i colle(fls from Tables of Obfervations continued for 82 years, that is from A. D. 1629 to 171 1, that the number of Births in London was, at a medium, about 14000 yearly : and like- wife, that the number of Males to that of Females, or the facility of their produdlion, is nearly as 18 to 17. But he thinks it the greatefl: weaknefs to draw any Argument from this againft the Influence of Chance in the produdion of the two Sexes. For, fays he, *' Let 14000 Dice, each having 35 faces, 18 white and 17 black, " be thrown up, and it is great Odds that the numbers of white and " black faces fhall come as near, or nearer, to each other, as the " numbers of Boys and Girls do in the Tables." To which the fhort anfwer is this: Dr. Arbuthnot never faid, " that fuppofing the facility of the production of a Male to that * See his two Letters to Mr. de Monmoti, one dated at London, 1 1 0£f. iyi2, the other horn Paris, 23 Jan. 1713, in the Appendix to thz Analyfe des Jeux de hazard, 2d Edit. " of TTse Doctrine oJ Chances. 253 " of the produdlion of a female to be already ^aY to nearly the Racio " of equality, or to that of j8 to 17 j he was amazed that the Ratio " of the numbers of Males and Females born fhould, for many years, " keep within fuch narrow bounds:" the only Propofition againft which Mr. Bernoulli'% reafoning has any force. But he might have faid, and we do ftill infift, that " as, from " the Obfervations, we can, with Mr. Bernou.'/i, infer the facili- " ties of produdlion of the two Sexes to be nearly in a Ratio of " equality J fo from this Ratio once difcovered, ^nd matiifejlly ferv- " ing to a "wife purpofe, we conclude the Ratio itfelf, or if you will *' the Form of' the Die, to be an Effect of Intelligence and Defgn'' As if we were fhewn a number of Dice, each with 18 white and 17 black, faces, which is Mr. Bernoulli's fuppofition, we fliould not doubt but that thofe Dice had been made by lonie Artift ; ar,d that their form was not owing to Chance, but was adapted to the particu- lar purpofe he had in Viev/. Thus much was neceffary to take off any imprefiion that the authority of fo great a name might make to the prejudice of our ar- gument. Which, after all, being level to the lowefi undcrffanding, and falling in with the common fenfe of mankind, needed no formal Demonftration, but for the fcholaftic fubtleties with vvliich it may be perplexed ; and for the abufe of certain words and piirafes ; which fometimes are imagined to have a meaning merely becaufe they are often uttered. Chance, as we underftand it, fuppofes the Exijlence of things, and their general known Properties : that a number of Dice, for inftance, being thrown, each of them flaail fettle upon one or other of its Bafes. After v/hich, the Probability of an affigned Chance, that is of fome particular difpofition of the Dice, becomes as proper a fub- j£(5l of Tnveftig.Ltion as any other quantity or Ratio can be. But Chance, in atheiftical writings or difcourfe, is a found ut- terly infignificant : It imports no determination to any mode of Ex- iflence ; nor indeed to Exijlence itfelf, more than to non-exiftcnce ; it can neither be defined nor underftood : nor can any Propofition' concerning it be either affirmed or denied, excepting this one, " That *' it is a mere word." The like may be faid of fome other words in frequent ufe ; as fate, necejjity, nature, a ccurfe of r,ature in contradiflindion to the Dii'lne energy : all which, as ufed on certain occafions, are mere founds : and yet, by artful management, they ferve to found fpe- cious conclufions : which however, as foon as the latent fdlacy of the Tertn is dettdted, appear to be no lefs abfurd in themfelves, than they commonly are hurtful to fociety. 1 fliaJL 254 ^^ Doctrine o/'Chances. I fliall only add, That this method of Reafoning may be ufefully ap- plied in fome other very interefting Enquiries ; if not to force the Aflent of others by a ftricft Demonftration, at lead to the Satisfa6tion of the Enquirer himfelf : and {hall conclude this Remark with a pafllige from the Ars CenjeBandi of Mr. James Bernoulli^ Part IV. Cap. 4. where that acute and judicious Writer thus introduceth his Solution of the Problem for AJfigning the Limits within ivhich, by the repetition of Experiments, the Probability of an Event may approach indefinitely to a Probability given, " Hoc igitur eft illud Problema &c." This^ fays he, is the Problem which I am now to impart to the Publick, after having kept it by me for twenty years : new it is, and difficult ; but of fuch excellent ufe, that it gives a high value and dignity to every other Branch of this Do£lrine. Yet there are Writers, of a Clafs indeed very different from that of fames Bernoulli, who infinuate as if the DcSlrine of Probabilities could have no place in any ferious Enquiry ; and that Studies of this kind, trivial and eafy as they be, rather difqualify a man for reafoning on every other fubjedl. Let the Reader chufe. PROBLEM LXXIV. To Jifid the Probability of throwing a Chance ajftgned a given number of times without intermijfion^ in any given number of Trials. Solution. Let the Probability of throwing the Chance in any one Trial be reprefentcd by ^ ^ , and the Probability of the contrary by — —7- : Suppofe n to reprefent the number of Trials given, and p the num- ber of times that the Chance is to come up without intermiffion ; then fuppofing — —r- = x, take the quotient of Unity divided by 1 — X — axx — aax"^ — a'^x'^ — a^x^ aP—^xP^ and having taken as many Terms of the Series refulting from that divifion, as there are Units in n — p + i> multiply the Sum of the P P P whole by ■ " " •■ , or by . " ■■ , and that Produd will exprefs bP a + i^P the Probability required. Exam- 'The Doctrine o/" Chances. 255 Example i. Let It be required to throw the Chance afligned three times toge- ther, in 10 trials, when« and b are in a ratio of Equality, otherwife when each of them is equal to Unity; then having divided i by 1 — X — XX — x% the Quotient continued to fo many Terms as there are Units in « — /" -h i> that is, in this Cafe to 10 — 3 -f- i = 8, will be I -J- X -]- 2XX -}- ^'^ + 7.^* + 13^5 4- 24x'5 \ 44x7. Where x being interpreted by — —7- , that is in this Cafe by — , the Series will become i-f-_|-i--fl_L-i--i.JJ._|_ 41 4- -i^ , of which the Sum is -^ = -^ , and this be- f p ing multiplied by ■■ " " , that is, in this Cafe by -^ , the ProducH: will be — ;y , and therefore 'tis fomething more than an equal Chance, that the Chance affigned will be thrown three times together fome time in 10 Trials, the Odds for it being 65 to 63. N. B. The continuation of the Terms of thofe Series is very eafy ; for in the Cafe of the prefent Problem, the Coefficient of any Term is the Sum of 3 of the preceding ; and in all Cafes, 'tis the Sum of fo many of the preceding Coefficients as are denoted by the num- ber/*. But if, in the foregoing Example, the ratio of ^ to ^ was of ine- quality, fuch as, for inftance 2 to i, then according to the prefcription given before, divide Unity by 1 — x — 2xx — 4^3, and the Qiio- tient v/ill be i -f- x -\-^xx -j- gx^ -\- igx'^^^gx^ -\- i22x'^ -\- 297^-7, in which the quantity x, which has univerfally been fuppofed : — ■ — -r-r- , will in this Cafe be = — j wherefore in the preceding Series having interpreted x by -^ , wc fhall find the Sum of 8 of its Terms will be ==: -^^ = — , and this being multiplied by — which in this Cafe is — ^ , the Produdl -^^ will exnrefs the Probability required, fo that there are the Odds of 592 to 137, that the Chance affigned will happen three times together in 10 Trials or before; and only the Odds of 41 to 40 that it does not happen three times together in j. Aftec- 2^'^ 7^5^ Doctrine o/ Chances. After having given the general Rule, it is proper to confider of Expedients to make the Calculation more eafy -, but before we pro- ceed, it is proper to take a new Cafe of this Problem : Suppofe there- fore it be required to find the Probability of throwing the Chance afligned 4 tin:ies together in 21 Trials. And firfl let us fuppofe the Chance aiTigned to be of J'quality, then we fhould begin to divide Unity by i — x — xx — x' — x* ; but if we confider that the Terms X 4- XX 4" -'>•■' + x^ ^^^ i" geometric Progreflion, and that the Sum of that Progreffion is ■ \_^^ - , if we fubtradt that flrom i, the remainder ■ ' ~ "_^7"' ^^'^ ^^ equivalent to i — x — xx — x'^ — x*, and confequently -j-zr^ l. y! will be equivalent to the Quotient of Unity divided by 1 — x — xx — Ar3 — x+ ; and therefore by that expedient, the moft complex Cafe of this Problem will be reduced to the contemplation of a Trinomial ; let us therefore begin to take fo many Terms of the Series refulting from the Divifion of Unity by the Trinomial i — 2a; 4- x' as there are Units in « — p -j- I, that is in 21 — 4 -j- i, or 18, and thofe Terms will be I -\- 2x -\- 4X* -\- Sx^ -{- i6x+ -\- 31x5 J^ 6ox' -\- ii6jf7 -\- 224A-^ -h 432a:9 -f 833^'° ■]- ',6c6x" -f 3096X" 4- 5968x'5 -\- ii494Ar''^ ~\- 22i55x'^ -j- 42704X'* + 823I2a:'7. Now al- though thefe Terms may feem at firft fight to be acquired by very great labour, yet if we confider what has been explained be- fore concerning the nature of a recurring Series, we fliall find that each Coefficient of the Series is generated from the double of the laft, liibtrafting once the Coefficient of that Term which ftands 5 places from the laft inclufive ; fo that for inftance if we wanted one Term more, confidering that the laft Coefficient is 82312, and that the Coefficient of that Term which ftands five places from the laft in- clufive is 5968, then the Coefficient required will be twice 82312, wanting once 5968, which will make it 158656, fo that the Term following the laft will be i58656x'^ But to make this more confpicuous, if we take the Binomial 2x — x\ and raife it fucceffively to the Powers, whofc Indices are o, i> 2, 3, 4, 5, 6. Gff. and add all thofe powers together, and write againft one another all the Terms which have the fame power of X, we ffiall have a very clear view of the quotient of i divided by I — 2x -\- xK Now it will fbnd thus, fuppofing that a ftands for 2. Ihe Doctrine of Chances, 257 I ■\-ax 4- a^x^ — 2ax^ -[- ax"^ ■ — • 3^^x7 -\- a'^x'^ 4fl'x* -ya'^x^ — 5a^x9 -1- a'°x'° — ea'x" -[- ix'" -\-a"x"— ya^x" -[- ^.ax" -{-a'^'x'^— Sa'^x'^ -\- ^ddx}"- -[-«"**:'* — i2fl"Ar'* 4-28«*;f'^ — 4tfx'« When the Terms have been difpofed in that manner, it will be eafy to fum them up by the help of a Theorem which may be feen pag. 224. Now a being ^= 2, and x =r — , every one of the Terms of the firft Column will be equal to i, and there- fore the Sum of the firft Column is fo many Units as there are Terms, which Sum confequently will be 18; but the Terms of the fecond Column being reduced to their proper Value, will con- ftitute the Series 32 I 32 • 32 I 32 • 32 ' 32 ' 32 ' 32 ' -2- + 22.. JL. _Li_ 4. -II. 4- _1L of which the Sum will be -^ : 32 • 32 ' 32 ' ?2 ' iZ 32 ' the Terms of the third Column will conftitute the Series — — + -J- 4- -^ 4. -i^ + -11- 4- -1L_ 4- _ii- 4- '^^"^ • I02d. I I02A I 102J. I \01± ' lO'l I 1074 < 1024 I 1024. ' 1024 I 1024 ' 10^4 ' 1024 ' 1024 of which the Sum is -— - j the Terms of the fourth Column added together are .^^g , and therefore the Sum of all Terms may be expreffed by is — -2i- 4- -^^ '-^ = -^^ . ' * ' 32 ' 1024 3*7^8 3270s L But 258 1})e Doctrine of Chances. But this Sum ought to be multiplied by i — at, that is, by I ^-■z=L — , which will make the Produft to be ' '^ - ^ - ' ■ ■ . Neverthelefs, this Multiplication by i — x, takes off too much from the true Sum, by one half of the loweft Term of each Column, therefore that half muft be added to the foregoing Sum ; now all the loweft Terms of each Column put together will be J _ _L1 + -± 11- == 221^ , of which the half ■"^"' 32 • 1024 32768 " 3279S 32708 ought to be added to the Sum . '^'^?^ l ■ , which will make the true y°'^^ ; but this is farther to be multiplied by which by reafon that a and b are in a ratio of equality will be reduced to xP ■==. -2— ; and therefore the Sum ■ ''^'° ' ought to be divided by 16, which will make it to be -j^-^— : and this lafl: Fradlion will denote the Probability of producing the Chance afligned 4 times fuccefTively fome time in 21 Trials, the Odds againft it being 527513 to 521063, which is about 82 to 81. But what is remarkable in this Problem is this, that the oftner the Chance afligned is to be produced I'ucceflively, the fewer Columns •will be neceflary to be ufed to have a fufficient Approximation, and in all high Cafes, it will be fufficient to ufe only tlic firft and fecond, or three at moft, whereof the firft is a geometric Progreflion, of which a very great number of Terms will be as eafily fummed up, as a very fmall number ; and the fecond Column by what we have faid concerning the nature of a recurring Series, as eafily as the firft, and in fliort all the Columns. But now 'tis time to confider the Cafe wherein a to b has a ratio of inequality ; we had faid before that in this Cafe we ought to divide Unity by i — x — axx — aax'i — a'x^ — af~^ xP ^ but. all the Terms after the firft which is i, conftitute a geometric Progreflion, of which the firft is x, and the laft ^z/— 'aV, and tiierefore the Sum of that Progreflion is — — 2-1. -, and this being fub- traded from Unity, the remainder will be i — ax •\-afxf+', and 1 • €IX therefore Unity being divided by the Series above-written will be 1 < — (IX ten 1 — ax lihe Doctrine o/'Chances. 259 I — ax and \i a A^ i be fuppofcd = w, this Fradlion I — ax -\- at xf-\r*^ — X will be J~°'' 3rr» ^^ therefore if we raife fuccefllvely mx — aPxP+^ to the feveral Powers denoted by o, i, 2, 3, 4, 5, 6, &c. and rank all thofe Powers in feveral Columns, and write againft one another all the Terms that have the fame power of x, we fhall be able to fum up every Column extended to the num- ber of Terms denoted by « — /> -\- j, which being done, the whole mufl: be multiplied by i — ax, and to the Sum is to be added the Sum of the loweft Term of each Column multiplied by ax. But if it be required to aflign what number of Games are ne- ceflary, in all Cafes, to make it an equal Chance whether or not f> Games will be won without intermilfion, it may be done by approximation, thus ; let -^-^ ZLl_ be fuppofed = q, and let •^ — " , _ be fuppofed = r, then the number of Games required will be expreffed by -^qr; thus fuppofing a -= 1, b ^=1, />=6, then the number of Games would be found between 86 and 875 but if a be fuppofed = i, and /^ = 2, ftill fuppofing ^ = 6, the number of Games requifite to that efFedt would be found to be between 763 and 764; but it is to be ob- ferved, that the greater the number p is, fo much the more exadl will the Solution prove. L 1 a Of TREATISE O F ANNUITIES O N LIVES: Dedicated to The Right Honourable GEORGE Earl of MACCLESFIELD President of the Royal Society.* PREFACE T O T H E Second EDITION. DR. Halley publifhed in the Philofophical Tranfadlions, N'>. 196. afi EJfay concerning the Valuation cf Lives ; it was partly built upon Jive Tears Obfervation of the Bills of Morta- lity taken at Breflaw, the Capital of Silefia, and partly on his own Calculation. Altho' 262 PREFACE. Althd' he had thereby confirmed the great Opinion "which the World entertained of his Skill arid Sagacity ^ yet he ivas fe7ifible, that his 'Ta- bles and Cakiilatiofis were fufceptible of farther Improvements ; of this he exprejjed his Senfe in the following Words ; Were this Calculus founded on the Experience of a very great Number of Years, it would very well be worth the while, to think, of Methods to facili- tate the Computation of two, three or more Lives. From whence it appears^ that the table of Obfervations bei?ig only the Refult of a few Tears Expedience ^ it was not fo entirely to be de- fended upon, as to make it the Foundation of a fixed and unalterable Valuation of Annuities on Lives; and that even admitting fuch a Ta- ble could be obtained, as might be grounded on the Experience of a great Number of Tears, flill the Method of applyifig it to the Valuation of feveral Lives, would be extremely laborious, confdering the vajl Num- ber of Operations, that would be requifite to combine every Tear of each Life with every Tear of all the other Lives. The SubjeSl of Annuities on Lives, had been long negleSied by me^ partly prevented by other Studies, partly wanting the necefary means to treat of it as it deferved: But two or three Tears after the Publica- tion oj the firfl Edition oj my Dodlrine of Chances, / took the SubjeB into Confideration ; and confulting Dr. HalleyV Table of Obfervations, I found that the Decrements of Life, for con fider able Intervals of Time, were in Arithmetic Progref/ion ; for Infiance, out of 6/^6 Perfons of twelve Tears of Age, there remain 640 after one Tear; 634 after two Tears ; 628, 622, 616, 610, 604, 598, 592, 586, after 3, 4, 5, 6, 7, 8, 9, \o Tears refpeBively, the common Difference of thofe Numbers being 6. Examining afterwards other Cafes, I found that the Decrements of Life for feveral Tears were fiill in Arithtnetic Progref/ion ; which may be obferved from the Age of 54, to the Age of ji, where the Difference for \j Tears together, is conftantly 10. After having thoroughly examined the Tables of Obfervation, and difcovered that Property of the Decrements of Life, 1 was inclined to compofe a Table of the Values cf Ainuities on Lives, by keeping clofe to the Tables of Obfervation ; which would have been done with Eafe, by taking in the whole Extent of Life, feveral Intervals whether equal or unequal : However, before I undertook the Tafk, f tried what would PREFACE. 263 would be the Refult, of Juppofing thofe Decrements uniform from the ^ge of Twelve ; being fatis fed that the Excejfes arifing on one fide, would be nearly compenfated by the DefeSts on the other ; then comparing my Cal- culation with that of Dr. Halley, 1 found the Conclufon fo very little different, that I thought it fuperfuous to join together feveral different Rules, in order to compofe a fingle one : I need not take notice, that from the Time of Birth to the Age of Twelve, the Probabilities of Life increafe, rather than decreafc, which is a Reafon of the apparent Irre- gularity of the Tables in the beginning. Another thing was necejjary to my Calcidation, which was, to fup- pofe the Extent of Life confined to a certain Period of Time, which J fuppofe to be at S6 : What induced me to affume that Suppofi- tion was ift. That Dr. Halley terminates his Tables of Obfervations at the 84''' Tear ; for alt ho' out of 1000 Children of one Tear of Age, there are twenty, who, according to Dr. Halley'^ Tables, attain to the Age of S4. Tears, this Number of zo is inconfiderable, and would fiill have been reduced, if the Obfervations had been carried two Tears farther. 2°. // appears from the Tables o/'Graunt, who printed the fir ft Edition of bis Book above 80 Tears ago, that out of \ 00 new-born Children, there remained not one after 86 Tears ; this was deduced from the Obfervations of feveral Tears, both in the City and the Coun- try, at a Time when the City being lefs populous, there was a greater Facility of coming at the Truth, than at prefent. 3°. I was farther confirmed in my Hypothefis, by Tables of Obfervation made in Switzer- land, about the Beginning of this Century, wherein the Limit of Life is placed at 86 ; As for what is alledged, that by fome Obfervations of late Tears, it appears, that Life is carried to 90, 95, and even to 100 Tears j I am no more moved by it, than by the Examples of Parr or Jenkins, the firfi of whom lived 152 Tears, and the other 167. To this may be added, that the Age for purchafing Annuities for Life, feldom exceeds -jq, at which Term^ Dr. Halley ends his Tables of the Valuation of Lives. The greatefi Difficulty that occurred to me in this Speculation, was to invent praSlical Rules that might eafily be applied to the Valuation of feveral Lives ; which, however, was happily overcome, the Rules being fo eafy, that by the Help of them, more can be performed in a garter of an Hour, than by any Method before extant^ in a garter of a Tear. Since 264 PREFACE. Since the Publication of my firjl Edition^ which was in lyz^, J made fame Improvements to it, as may be feen in the j'econd Edition of my Doftrine of Chances ; but this Edition of the Annuities has many Advantages over the former^ and that in refpeSi to the Dijpofition of the Precepts, the Concifenefs of the Rules, the Multiplicity of Problems^ and Uftfulnefs of the Tables I have invented. Before I make an End of this Preface, I think it proper to obferve, that alt ho' I have given Rules for finding the Value of Aniiuities for any Rate of Inter eft, yet I have co77fined myfelf in my Tables, to the fe- ver al Rates of /[, ^ and t per Cent, which may be interpreted, as if I thought it reafonable, that when Land fcarce produces three and a half per Cent, and South-Sea Annuities barely that Intereft, yet the Pur chafer of an Annuity f mild make 4 per Cent, or above-, but thofe Cafes can hardly admit of Comparifon, it being well known, that Land in Fee-Jimple procures to the Proprietor Credit, Honour, Reputation, and other Advantages, in confideration of which, he is contented with a f ma Her Income. As to the Value of South- Sea Annuities, it has its Foundation on the PunBuality of Payments, and on a Parliamentary Security ; but Ammities on Lives, have not the former Security, and feldom the latter. It was found neceflary, however, in a fubfequent Edition, to add the Tables of 3 and 'i^\ per Cent, Intereft. ANNUI- Of ANNUITIES on LIVES. Part I. containing the Rules and Examples. BEFORE I come to the Solution of Queftions on Lives, it will be neceflary to explain the Meaning of fome Words which I fhall often have occafion to mention. 1°. Suppofing the Probabilities of Life to decreafe in Arith- metic Progreflion in fuch manner, as that I'uppoling, for In- flance, 36 Perfons each of the Age of 50, if after one Year ex- pired there remain but 35, after two 34, after three 33, and fo on ; it is very plain that fuch Lives would neceflarily be extindt in 36 Years, and that therefore the Probabilities of living i, 2, 3, 4, 5, t^c. Years from this Age of 50 would fitly be reprefented by the Fraftions — , -\j- , -^ , -^ , ^ , (^c. which decreafe in Arith- metic Progreffion. I will not fay that the Decrements of Life are precifely in that Proportion ; ftill comparing that Hypothefis with the Table of Dr. Halley, from the Obfervations made at Brejlaw, they will be found to be exceedingly approaching. 2°. I call that the Complement of Life, which remains from the Age given, to the Time of the Extindion of Life, which will be at 86, according to our Hypothefis. Thus fuppofing an Age of 50, becaufe the Difference between 50 and 86 is 36, I call 36 the Complement of Life. 3°. I call that the Rate of Interefl which is properly the Amount of one Pound, put out at Interefl for one Year ; otherwife one Pound joined with the Interefl it produces in one Year : thus fuppofing Interefl at 5 per Cent the Interefl of i /. would be 0.05, which baing joined to the Principal i, produces 1.05; which is what I call the Rate of Interefl. PROBLEM L Suppofing the Probabilities of Life to decreafe in Arith- metic ProgreJJion, to find the Value of an Aufiuity upon a Life of an Age given. Solution. Let the Rent or Annuity be fuppofed=i, theRateof Interefl = r, the Complement ofLife=;/, the Value of an Annuity certain to continue M m during 266 n^e Doctrine of Chances applied r ■ \--P during « Years =P, then will the Value of the Life be -;-—' ^^^'"^^ is thus expreffed in Words at length ; Take the Value of an Annuity certain for fo many Tears, as are denoted by the Complement of Life ; multiply this Value by the Rate of Interejl, and divide the Product by the Complement of Life, then let the ^lotient be fubtraSfcd from Unity, and let the Remainder be di- vided by the Inter eft oj i 1. then this lajl ^cotient •will exprefs the Value of an Amiuity for the Age given. Thus fuppofe it were required to find the prefent Value of an Annuity of i /. for an Age of 50, Intereft being at 5 per Cent. The Complement of Life being 36, let the Value of an Annuity certain, according to the given Rate of Intereft, be taken out of the Tables annexed to this Book, this Value wilt be found to be 16.5468. Let this Value be multiplied by the Rate of Intereft 1.05, the Produdt will be 17.374.1. Let this Produdt be divided by the Complement of Life, viz. by 36, the Quotient will be 0.4826. Subtradl this Quotient from Unity, the Remainder will be 0,5174. Laftly, divide this Quotient by the Intereft of 1 /. viz. by 0.05, and the new Quotient will be 1035; which will exprefs the Va- lue of an Annuity of i /. or how many Years Purchafe the faid Life of 50 is worth. And in the fame manner, if Intereft of Money was at 6 per Cent. an Annuity upon an Age of 50, would be found worth 9.49 Years Purchafe. But as I have annexed to this Treatife the Values of Annuities for an Intereft of 3, 3 1, 4, 5, and 6 per Cent, it will not be neceflary to calculate thofe Cafes, but fuch only as require a Rate of Intereft higher or lower, or intermediate ; which will feldom happen, but in cafe it does, the Rule may eafily be applied. PROBLEM II. 77}e Values of twoftngle Lives being given, to fnd the Va- lue of an Annuity granted for the Time of their joint continuance* SOLU- to the Valuation ^Annuities. 267 Solution. Let M be the Value of one Life, P the Value of the other, r the Rate of Intereft ; then the Value of an Annuity upon the two joint Lives will be - — ^ — — ^ — ^ , in Words thus ; Multiply together the Values of the two Lives, and referve the ProduB. Let that ProdiiB be again tnultiplied by the Intereft of i\. and let that new ProduSl be fubtraSiedfrom the Sum of the Values of the Lives, and referve the Remainder. Divide the fir ft ^antity referved by. the fecond, and the ^otient will exprefs the Value of the two joint Lives. Thus, fuppofing one Life of 40 Years of Age, the other of 50, and Intereft at 5 per Cent. The Value of the firft Life will be found in the Tables to be 1 1.83, the Value of the fecond 10.35, ^^^ P^°' dudl will be 1 22.4405, which Producft muft be referved. Multiply this again by the Intereft of i /. viz. by 0.05, and this new Produdl will be 6.122025. This new Produdl being fubtrafted from the Sum of the Lives which is 22.18, the Remainder will be 16.057975, and this is the fecond Quantity referved. Now dividing the firft Quantity referved by the fecond, the Quo- tient will be 7.62 nearly j and this exprefles the Values of the two joint L.ives. If the Lives are equal, the Canon for the Value of the joint Lives will be ftiortened and be reduced to — ==-— t7 > which in words may be thus exprefTed ; Take the Value of one Life, and referve that Value. Multiply this Value by the hit cr eft of 1 1. and then fabtraSi the Pro- duSf from the Number 2, and referve the Remainder. Divide the firft ^.antity referved by the fecond, a id the ^otient will exprefs the Value of the two equal joint Lives. Thus, fuppofing each Life to be 45 Years of Age, and Intereft at 5 per Cent. The Value of one Life will be found to be 11.14, the firft Quan- tity referved. This being multiplied by 0.05 the Intereft of i /. the Produdl will be 0.557. This Produft being fubtraded from the Number 2, the Re- mainder will be 1.443, ^he fecond Quantity referved. M m 2 Divide 268 The Doctrine of Chances applied Divide the firft Quantity referved viz. 11.14-, by the fecond, •y/z. 1.443, and the Quotient 7.72 will be the Value of the two joint Lives, each of 45 Years of Age. PROBLEM III. The Values of three /ingle Lives being given^ to ji?id the Value of an Annuity for the Ti??ie of their joint co?i- ti?mance. Solution. Let M, P, ^, be the refpeiflive Values of the fingle Lives, then the Value of the three joint Lives will be MP+M^^t^^-zj ^u^^^ ' fuppofing d to reprefent the Intereft of 1 /. in words thus ; Multifly the Values of the fingle Lives together, and referve the Produdf. Let that ProduSi be multiplied again by the Interejl of 1 1. and let the Double of that new ProduB be fubtraSfed from the Sum ofthefe- veral ProduBs of the Lives taken two and two, and referve the Re- mainder. Divide the firft ^lantity referved by the fecond, and the ^otient will be the Value of the three joint Lives. Thus, fuppofing one Life to be worth i 3 Years Purchafe, the fecond 1 4, the third 1 5, and Intereft at 4 per Cent, the ProducH: of the three Lives will be 2730, which being multiplied by the Intereft of 1 /. viz. by 0.04, the newProdudl will be 109.20, whereof the double is 2 1 8.^ : Now the Produd of the firft Life by the fe- cond is I 82 ; the Produdt of the firft Life by the third is 195; and the Produdl of the fecond Life by the third is 210, the Sum of all which is 587; from which fubtrading the Number 218.40 found above, the Remainder will be 368.60, by which the Produd of the three Lives, viz. 2730 being divided, the Quotient 7.41 will be the Value of the three joint Lives. But if the three Lives were equal, the general Expreftion of the Value of the joint Lives will be much ftiorter : for let Mreprefent the Value of one Life, d the Intereft of i /. then the Value of the three joint Lives will be pr^j^j , in Words thus j Take to the Valuation (?/'Annuities. 269 T^ake the Value of one Life, and referve it, multiply this Value by the Inter eji of i\. and double the ProduSl. Subtract thii double ProduSi from the number 3, and referve the "Remainder. Divide the firjl ^antity referred by the fecond, and the S^otient ivill be the Value of the three joint Lives. Thus, fuppofirig three equal Lives each worth 14 Years Pur- chafe, referve the Number 14. Multiply this by 0.04, Intereft of i /. the Produdt will be 0.56, which being doubled, will be 1.12. This being fubtradted from the Number 3, the Remainder will be 1.88, which is the fecond Quantity to be referved. Divide 14, the firft Quantity referved by the fecond 1.88, and the Quotient 7.44 will be the Value of the three joint Lives. From the two laft Examples it appears, that in eftimating the Va- lues of joint Lives, it would be an Error to fuppofe that they might be reduced to an Equality, by taking a Mean Life betwixt the longed and fliorteft, for altho' 14 is a Medium betwixt 13 and 15, yet an Annuity upon thofe three joint Lives was found to be 7.41, whereas fuppofing them to be each 14 Years Purchafe, the Value is 7.44 ; it is true that the Difference is fo fmall, that it might be negledted, yet this arifes meerly from a near Equality in the Lives ; for if there had been a greater Inequality, the Conclufion would have confidera- bly varied. Before I come to the fourth Problem, I think it proper to ex- plain the Meaning of fome Notations which I make ufc of, in order to be as clear and concife as I can. I denote the Value of an Annuity upon two joint Lives, whofe fingle Values are M and P hy M P, which ought carefully to be diftinguifhed from the Notation MP ; this laft denoting barely the Produ£l of one Value multiplied by the other, whereas MP ftands for what was denoted in our fecond Problem by MP Ai-rP—r—iMf In the fame manner, the Value of an Annuity upon the three joint Lives whofe fingle Values are M, P, ^, is denoted by AIP^^,, which is equivalent to what has been expreffed in the tliird Problem by- iMP^ A/P-j- M^-t Pii~ zd M P^!_ • This being premlfed, I proceed to the fourth Problem. PRO^ 270 T^he Doctrine of Chances applied PROBLEM IV. Tloe Values of two ftngle Lives being given, to find the Value of an A?inuity upoji the longefl of them^ that is, to continue fo long as either of them is in being. Solution. Let M be the Value of one Life, P the Value of the other, MP the Value of the two joint Lives, then the Value of the longefl of the two Lives will be M-\-P~MF. In Words thus ; From the Sum of the Values of the fingle Lives, fubtraSl the Value of the joint Lives, and the Remainder ivill be the Value of the longeft. Let us fuppofe two Lives, one worth 13 Years Purchafe, the other 14, and Intereft at i^ per Cent. The Sum of the Values of the Lives is 27, the Value of the two joint Lives by the Rules before given, will be found 9.23. Now, fubtrading 9.23 from 27, the Remainder 17.77 is the Value of the longeft of the two Lives. If the two Lives are equal, the Operation will be fomething fliorter. But it is proper to obferve in this place, that if feveral equal Lives are concerned in an Annuity, I commonly denote one fingle Life by M', two joint Lives by M" , three joint Lives by M'" , and fo on ; fo that the Rule for an Annuity to be granted till fuch Time as either of the equal Lives is in being may be exprefTed by ^M.' — M" . PROBLEM V. "The Values of three fingle Lives being given^ to find the Value of a?i Anfiuity upon the longefl of them. SOLUTI ON. Let M, P, ^, be the Values of the fingle Lives, MP, M^, P^, the Values of all the joint Lives combined two and two, MP^ the Value of three joint Lives, then the Value of an Annuity upon the longefl of them is M-\-P-\- ^—MP- M^—PZrh^^P^, in Words thus ; Take the Sum of the three fingle Lives, Jrom 'which Swn fubtraB the Sum of all the joint Lives combined two and two, then to the Remainder add the Value of the three joint Lives, and the Refult will be the Value of the longeft of the three Lives. Thus, to the Valuation (?/ Annuities. 271 Thus, Suppofing the fingle Lives to be 73, 14, and 15 Years Purchafe, the Sum of the Values will be 42 ; the Values of the firft and fecond joint Lives is 9.24, of the firft and third 9.05, of the fecond and third 10.18, the Sum of all which is 29.06 which being fubtradted from the Sum of the Lives found before, viz. 42, the Remainder will be 12.94, to which adding the Value of the three ioint Lives 7.41, the Sum 20.35 will be the Value of the longefi of the three joint Lives. But if the three Lives are equal, the Rule for the Value of the Life that remains laft is 3 M' — 3 M" ^ M'" . Of REVERSIONS. PROBLEM VL Suppofe A is in Pojfejfwn of an Annuity^ and that B af- ter the Deceafe of A is to have the Annuity for him^ and his Heirs for ever^ to find the prefent Value of the Reverfon. Solution. Let M be the Value of the Life in PofTeffion, r the Rate of In- tereft, then the prefent Value will be — ^- M, that is, from the Value of the Perpetuity, fubtraB the Value of the Life in Poffejion, and the Remainder will be the Value of the Re'ucrjion. Thus, Suppofing that A is 50 Years of Age, an Annuity upon his Life, Intereft at 5 per Cent, would be S.39, which being fub- tradled from the Perpetuity 20, the Remainder will be 11.61, which is the prefent Value of the Expectation of B. In the fame manner, fuppofing that C were to have an Annuity for him and his Heirs for ever, after the Lives of A and B, then from the Perpetuity fubtrading the Value of the longeft of the two Lives of A and B, the Remainder will exprefs the Value of C's Expectation. Thus, Suppofing the Ages of y^ and 5 be 40 and 50, the Value of an Annuity upon the longeft of thefe two Lives would be found by the4*'' Problem to be i 4.56 ; and this being fubtradled from the Per- petuity 20, the Remainder is 5.44, which is the Value of C'sEx- pedtation, and the Rule will be the fame in any other Cafe that may be propofed. PRO^ 272 The Doctrine of Chances applied PROBLEM VII. Suppojing that A is in Pojfejfton of an Annuity for his Life^ and that B after the Life of h^ JJjould have an Annuity for his Life only ; to fnd the Value of the Life of B after the Life of A. This Cafe ought carefully to be diftinguiftied from the Cafe of the 6th Problem ; for in that Problem, altho' the Expedtant B fhould die before A., flill the Heirs of B have the Reverfion ; but in the Cafe of the prefent Problem, if B dies before Ay the Heirs of B have no Expe<ftation. Solution. Let M be the Value of the Life of the prefent. Pofleflbr, P the Value of the Life of the Expedlant, then the Value of his Expefta- tion is P — MP. In Words thus ; From the prefent Value oj the Life of B, fiibtraSl the prefent Value of the joint Lives of B and A, and the Remainder will be the Value of B'5 ExpeSiation. The Reafon of which Operation is very plain, for if B were now to begin to receive the Annuity, it would be worth to him the Sum P in prefent Value ; but as lie is to receive nothing during the joint Lives of himfelf and A, the prefent Value of their two joint Lives ought to be fubtraded from the Value of his own Life. PROBLEM VIIL To find the Value of one Life after two. Thus^ Suppofe A in PoffeJJion of an A?muity for his Life., that B is to have his Life in it after A, and that C is likewife to have his Life in it after B, but fo that B dying before A, Cfucceeds A immediately ; to find the Value of Cs Expeciation. Solution. Let M, P, ^, be the refpedtive Values of the Lives of A, B, C, then to the Valuation 0/ Annuities. 273 then the Value of Cs Expcdation is ^— ^^4-^-^^ which in Words is thus exprefled -, From the prefent Value of the Life of C, fubtraB the Sum of the joint Lives of himfelf and A, and of himfelf and B, and to the Re- mainder add the Sum of the three joint Lives, and the Refult of thefe Operations will exprefs the prefent Value of the ExpeSlation ofC. PROBLEM IX. If Ay B, C agree among themf elves to buy an Annuity to be by them equally divided^ whiljl they live together, then after the Deceafe of one of them, to be equally di- vided between the two Survivor s^ then to belong entire- ly to the lafl Survivor for his Life ; to find what each of them ought to contribute towards the Purchafe. Solution. Let M, P, ^, be the refpedlive Values of the Lives of y/, P, C, then what A is to contribute, is What B is to contribute, is What C is to contribute, is In Words thus ; From the Value of the Life of h, fubtradl the ha f Sum of the Values of the joint Lives of himfelf and B, and of himfelf and C', and to the Remainder add | of the Value oj the three joint Lives, and the Sum ivill be what A is to contribute towards the Purchafe. In like manner, from the Value of P's Life fubtradl the half Sum of the Values of the joint Lives of himfelf and A, and of himfelf and C, and to the Remainder add | of the Value of the three joint Lives, and the Sum will be what B is to contribute. RI n And 2 74 T'^^ Doctrine of Chances applied And again, from the Value of the Life of C, fubtrad the half Sum of the V^alues of himfelf and A, and of himfelf and B ; then to the Remainder add | of the Values of the three joint Lives, and this laft Operation will ftiew what C is to contribute. PROBLEM X. Suppojing three equal Lives of any Age given ^ for Inflance 30, and that up07i the FaiVmg of any one of them^ that Life pall be immediately replaced^ and I then receive a Sum 1 agreed upon^ and that to Perpetuity for me and my Heirs j what is the prefent Value of that ExpeEia- tion^ and at what Intervals of Time, one with another .y may I expeB to receive the f aid Sum? Solution. Imagine that there is an Annuity of i /, to be received as long as the three Lives are in being, and that the Prefent Value is M"\ which Symbol we make ule of to reprefent the Prefent Value of an Annuity upon three equal joint Lives ; now, fince each Life is fuppofed to be 30 Years of Age, and that the Rate of Intereft is 5 per Cent, we fhall find, by following the Direftions given in Prob. IIL that the Pre- fent Value of the three joint Lives is 7.64 = M'" j this being fixed, the Prefent Value of all the Payments to be made to Eternity at equal Intervals of Time, will be -'~^,,, — x^ where the Qiiantity d figni- fies the Intercft of i /. In words thus ; Multiply the Prefent Value of the three joint Lives, viz. 7.64, by the Iiitereft of 1 1. u-hich in this Cafe is 0.05, and that Produ^, which is 0.382, nwjl be referved. SubtraB this ^antity jrom Unity, and the Remainder, viz. 0.6 J 8 being divided by the ^antity referved, the ^otient will be 1.62, and this being multiplied by the Sum f, which we may fuppofe \ 00 1, the ProduSi will be 162 1. and this is the prefent Value of all the Payments that will be made to Eternity , at equal Intervals of time upon the fail- ing of a Lije, which is to be immediately replaced. As for the Intervals of Time after which thofe Replacements will be made, they may be found thus ; Look to the Valuation ©/"Annuities. 275 Look in the feventh of our Tables for the Number 7.64, which is the Value of the three joint Lives, and over againft it will be found the Number anfwering, which is between 9 and 10 ; and fo it may be faid that the Replacements will be made at every Interval of about 9 or I o Years. But that Interval may be determined a little more accurately, by help of a Table of Logarithms, by taking the Logarithm of the Quantity - _^^'" ' and dividing it by the Logarithm of r. The Logarithm of ^_j^^ , is 0.20901 15 ; the Logarithm of r is 0.021 1893 ; and the firfl being divided by the fecond, the Quo- tient is 9.86, which fhews that the Replacements will be made at Intervals a little more than 9 \ Years. PROBLEM XL Suppofing^ as before^ three equal Lives <^ 30, and that the Lives are 720 1 to be renewed ^ till after the failing of any two of them^ and that a Sum p is then to be re- ceived^ and that perpetually^ after the failing of two Lives^ what is the prefent Value of that ExpeSiation f Solution. Make 3 M" — 2 M"' =1 A, let the Intereft of i /. be = d, then the prefent Value of that Expedtation will be - '~^' — x p. But to know the Intervals of Time after which the Lives will be filled up, take the Logarithm of the Quantity ^ ' — , and divide it by the Logarithm of r. The Value of a fingle Life of 30, Intereft at 5 per Cent, is found in our Tables to be 13 Years Purchafe, the Value M" of two joint Lives by Problem 11. is 9.63 ; and the Value M'" of three joint Lives by Problem III. is 7.64 ; then 3 M" — zM'" , or the Difference be- tween the Triple of two joint Lives, and the Double of three joint Lives will be 13,59 =:yf, then —j^ X/ will be found to be °-473 /*' and the Intervals of Time will be 23.32, that is, nearly 23^ Years. N n 2 PRO- 276 ^Jhe Doctrine of Chances applied PROBLEM XII. Suppoftng Jim the Lives to be 30, avd that they are not to be renewed till after the ExtinElion of all three ^ and that a Sufn q is then to be received^ and that perpetually after every RetiewaU what is the prefent Value of that ExpeBation f Solution. Make 3M' — 3M" -j- M"' = B, then the prefent Value of that Expedation will be — j^— xg; here B will be found to be 17.76, and confequently — j^ — x g will be = o. 12 1 xq. And the Intervals of Time will be the Logarithm of the Quantity • _' divided by the Logarithm of r, which in this Cafe would be 44.87, that is, nearly 45 Years. Corollary. Hence it will be eafy for the Proprietor of the Lives, to find which is moft advantageous to him, to fill up a Life as foon as i't is vacant, or not to fill up before the Vacancy of two, or to let them all drop before the Renewal. Remark. It is not to be imagined that if Intereft of Money was higher or lower than 5 per Cent, the Intervals of Time after which the Re- newals are made, would be the fame as they are now, for it will be found, that as Intereft is higher, the Intervals will be fhorter ; and as it is lower, fo the Intervals will be longer ; yet one might make it an Objedion to our Rules, that the length of Life would thereby feem to depend upon the Rate of Intereft. To anfwer this Diffi- culty, it mufl be obferved, that the calculating of Time imports no more, than that confidering the Circumflances of the Purchafcr and the Proprietor of the Lives, in refpedl to the Rate of Interefl agreed upon, and the Sum to be given upon the Renewal of a Life, or Lives, the Proprietor makes the fame Advantage of his Money, as if he had agreed with the Purchafer, that he fliould pay him a cer- tain Sum of Money at equal Intervals of Time, for redeeming the Rifque tll'lt to the Valuation o/" Annuities. 277 Rifque which he the Purchafer runs of paying that Sum when the Life or Lives drop : but the real Litervals of Time will be (hewn afterwards. Altho' it feldom happens that in Contrads about Lives, any more than three are concerned, yet I hope it will not be difpleafing to our Readers to have this Speculation carried a little farther. But as general Rules are beft inculcated by particular Examples, I fhall take the Cafe of five Lives, and exprefs the feveral Circum- flances of them in fuch manner, as that they may be a fure Guide in all other Cafes of the fame kind, let the Number of Lives be what it will ; let therefore the following Expreflions be written, SM""-ArM"" ioM"'—\sM""-\-6M' 10 M" — zoM"'-X-isM"" — 4M' SM'--xoM" -\-ioM"'^ sM""-\-iM' The firft Term M'"' reprefents properly the prefent Value of an Annuity upon five equal joint Lives, but from ihence may be de- duced the Time of their joint continuance, or the Time in which it may be cxpedled that one of them will fail, it being as I have faid before, the Logarithm of _^^yy ■ divided by the Logarithm of r : however, for fhortnefs fake, I call for the prefent that Expreffion the Time. The two next Terms, 5M"" — 4M'"", reprefent the Time in which two of the Lives will fail. The three next Terms, \oM"' — -i^M"" -\- bM"'\ reprefent the Time in which three out of the five Lives will fail. The four next, i o M" — 20 M'" + 1 5 M"" — 4 M"'\ reprefer.t the Time in which four out of the five Lives will fail. The five next, 5M'— 10 M"-f 10 M'" — 5 M'"+ i M'"", re- prefent the Time in which all the five Lives will be extinil. Now the Law of the Generation of the Co-efficients is thus. 1°. Take all the Terms which are afi:e(fted with the Mark M""\ beginning from the uppermoft, with the Co-efiicients i — 4 -h 6 — 4 -f- 1 , which are the Terms of the Binomial i — i , raifed to the fourth . Power, which is lefs by one than the Number of Lives concerned. 2". Take the Terms which are affected with the Mark M"\ and prefix to them in order, the product of the Number 5 by the Co- efficients 1 — 3-I-3 — 1> w'hich are the Terms of the Bbomial 1 — i raifed to its Cube, that is, to a Power lefs by two than the Number of Lives concerned. 3"- 2yS 72>e Doctrine of Chances applied 3°. Take all the Terms which are affeded with the Mark M"\ and prefix to them in order, the Product of the Number lo, mul- tiplied by the Co-efficients i — 2-|-i, which are the Terms of the Binomial i — i raifed to its Square, that is, to a Power lefs by three than the Number of Lives concerned. 4°. Take all the Terms which are affedted with the Mark M'\ and prefix to them the product of the Number lo, multiplied by the Terms of the Binomial i — i, raifed to the Power whofe Index is i, that is to a Power lefs by four than the Number of Lives con- cerned. 5°. Take all the Terms which are affedted with the Mark M\ and prefix to them the Product of the Number 5, multiplied by the Binomial i — i, raifed to a Power lefs by 5 than the Number of Lives concerned ; which in this Cafe happening to be nothing, or o, dege- generates barely into Unity. As for the Multiplicators, conceiving that the Multiplicator of the firfl: Term M'"" is 1, all the Multiplicators will be i, 5, 10, 10, 5, which are all, except the laft, the Coefficients of the Binomial i -\- 1, raifed to its fifth Power, that is, to a Power equalling the Number of all the Lives. N. B. The Exception here given, does not fall upon the Number 5, but upon the laft Term of the fifth Power, i-j-5-i-io-^io-l-5-|-J, which laft i is rejected. Of SUCCESSIVE LIVES. PROBLEM Xin. If A enjoys an Annuity for his Life., and at his Deceafe has the Nomination of a Succeffor B, ijoho is alfo to efijoy the Annuity for his Life, to fnd the prejent Value of the two fucceffive Lives. Sol u T ION. Let the Values of the Lives be M and P ; let i/ be the Intereft of I /. then the Value of the two fucceffive Lives will be M-\-P — dMP. But if the Succeflbr B was himfelf to have the Nomination of a Life ^; then the Value of t he three fucceffive Lives would by M-^-P-}- But to the Valuation ^Annuities. 279 But before I proceed, it is proper to obferve that the Expreffions MP, M^, P^, and MP^, fignify barely Produdts, which is conformable to the ufual Algebraic Notation ; this I take notice of, for fear thofe Expreffions fhould be conf ound ed with others that I have made ufe of before, viz. MP, M^, P^, and MP^, which denoted joint Lives. But to comprife under one general Rule all the poffible Cafes that may happen about any Number of fucceffive Lives, it will be proper to exprefs it in Words at length, thus ; From the Sum of all the Lives, fiibtro6l the Sum of the PrcduBs of all the Lives com billed two and two, which Sum of ProdiiSfs before they are fubtr-'Sted, ought to be multiplied by the Inter eji of \\. 'to this add the Sum of the ProduSls oj all the Lives taken three and three, but multiplied again by the Square of the Inter e[i of 1 1. From thisfubtraSi the Sum of the Products of all the Lives taken four and four, but mu'tiplied again by the Cube of the Inter e/i of \\. andfo on by alternate Additions and Subtractions ftill cbferving that if there was occafwn to take the Lives five and five, fix and fix, &c. the Intereji of 1 1. ought to be raifed to the /^th Power, and to the ^th, andfo on. But all thofe Operations would be very much contrafted, if the Lives to be nominated were always of the fame Age, for Inftance 30 : for fuppofe Mto be the Value of an Annuity on an Age of 30, and d to be the Intereft of i /. then the prefent Value of allthe fuc- ceffive Lives, of which the Number is n, would be ——~-j-— • In Words thus ; Multiply the Value of one Life by the Inter efl of i\. let the ProduB be jubtr acted from Unity, ajid let the Remainder be raifed to that Power which anfwers to the Number of Lives ; then this Power being again fubtraSled from Unity., let the Remainder be divided by the Inte- reft of \\. and the ^totientwill be the prefent Value of all the Juccejive equal Lives. And againy ij the Number of thofe Lives were infinite, the Sum. would barely be — . PROBLEM XIV. Of a Perpetual Advowfon. 1°. r fuppofe that at the Time of the Demife of the Incumbent, the Patron would receive the Sum f, for alienating his Right of the next 2 8o The Doctrine of Chances applied next Prefentation, if the Law did not forbid the Alienation in that Circumftance of Time. 2°. I fiinpofe that when this Right is transferred, the Age of the Incumbent is fuch, that an Annuity upon his Life would be worth M Years Purcliafe, when the Intereft of i /. is d. This being fuppofed, the Right of the next Prefentation is worth I — d'M\f, and the Right of Patronage, or perpetual Recurrency of the like Circumftances to Eternity, would be worth ' ~ — yif. In words thus; 'Take the pre fen t Value of the Life of the Incumbent, and multiply it by the Inter ejl of i\. and referve the ProduSi. SubtraSl this PreduBfrom Unity, and let the Remainder be multiplied by the Sum expeffedf, and the new Producl iinll Jkew the Right of the next Prefentation ; let atfo this be referred. 'Then divide tij'fecojid ^lantity referved by the fir fl, and the patient nvill Jloeio the prefent Value of the Right of Patronage, or perpetual Re- currency. Thus, fuppofing the Life of the Incumbent worth 8 Years Pur- chafe, the Rate of Intereft 5 per Cent, and the Sum / to be 100/. tlie Right of the next Prefentation would be worth 60/. and the Right of perpetual Recurrency 150 /. PROBLEM XV. Of a Copy- /jo Id. Stippofing that every Copy -hold Tenant pays to the Lord of the Manor a certain Fine on Admittance^ and that every Succeffor does the like ; to find the Value of the Copy -hold computed from the Time of a Fine being paid^ independently fro?n the Fine that may be given on Alie- nation. Solution. I fuppofe that the Value of the Life of the prefent Tenant, and the Life of every future SuccclTor when he comes to Pofleflion is the fame ; this being admitted, let M be the Value of a Life, d the In- tereft of I /. and /the Fine to be paid, then the prefent Value of the Copy- to the Valuation of Annuities. 281 Copy-hold will be — ^J xf: and this Expreflion being exactly the fame as that whereby the Right of Patronage has been deter- mined, needs no Explanation in Words. Only it is neceffary to obferve, that the Sum / paid in Hand being added to this, will make the Canon fliorter, and will be reduced to -j-jj^ , which may be exprefled thus in Words. Divide the Fine by the ProduSl of the Lije, multiplied by the Inter efi ofi\. Thus, if the Life of a Tenant is worth 12 Years Purchafe, and the Fine to be paid on Admittance 56 /. and alfo the Rate of In- tereft 5 per Cent, then the prefent Value of the Copy-hold is 93 u. 3 PROBLEM XVI. A borrows a certain Sum of Money, and gives Security that it Jhall be repaid at his Deceafe with the Interefis ; 40 fx the Sum which is then to be paid. Solution. Let the Sum borrowed be / the Life of the Borrower M Years purchafe, d the Intercft of i /. then the Sum to be paid at A\ De- ceafe will be ^_!_jiyi ; thus, fuppofing /= 800, Af = 11.83, </=o.o5, then ^_^^^ would be found = 1958/; in the fame manner, if the Sum to be paid at A'% Deceafe, was to be an Equi- valent for his Life, unpaid at the Time of the Purchafe, that Sum would be ^__^^^ = 2895/. Supppofing the Annuity received to be 100/. as alfo the Life of ^ 1 1.83 Years Purchafe. PROBLEM XVIL A borrows a Sum f, payable at his Deceafe, but with this Condition, that if he dies before B, then the whole Sum is to be loft to the Lender ; to find what A ought to pay at his Deceafe in cafe hefurvives B. O o S O L u- 282 The Doctrine of Chances applied Solution. Let us fuppofe, as before, that A is 40 Years of Age, that the Sum borrowed is 800 /. and that Intereft of Money is 5 per Cent. Farther, let it be fuppofed that B is 70 Years of Age, then, 1°. determine what A fhould pay at his Deceafe, if the Life of B was not concerned ; by the Solution of the preceding Problem, we find the Sum to be 1958/. But we ought to confider that the Lender having a Chance to lofe his Money, there ought to be a Compenfation for the Rifque he runs, which is founded on the poffibility of a Man of feventy outlif ing a Man of forty. Now, by the Rules to be deli- vered in the next Problem, we (hall find that the Probability of that Contingency is meafured by the Fraftion -^ , and therefore the Pro- bability of the youngeft Life's furviving the oldeft is — . Now this being the Meafure of the Probability which the Lender has of being repaid, the Sum 1958 ought to be increafed in the proportion of 23 to 19, which will make it to be 2370/. nearly. Of the Probabilities of Survivorjhip, PROBLEM XVIII. Any Number of Lives being given^ to find their Probabi- lity of Survivorjhip. Solution. Let A^ B, C, D, C^c. be the Lives, whereof A is fuppofed to be the youngeft, B the next to it, C the next, &c. and fo the laft the oldeft. Let w, p, q, s, /, &c. be the refpedlive Intervals intercepted be- tween the Ages of thofe Lives, and the Extremity of old Age fup- pofed at 86 ; then the Probabilities of any one of thofe Lives fur- viving all the reft, will be for to the Valtmtion o/'Amnuities. 283 ior A 1 — I p n zn bnp P 11 i ' t* B IZ nfq J5 ZOnpq s c i^ zn tnp r ^„p — I inpq ZOtipqs D iznfq 20 npqt E ^npq . 1 zonpqt &c. T^ S"P1' Here fome few things may be obferved. 1°. That the Probability of the youngeft Lifer furviving all the reft, always begins with Unity, and that it is expreffed by fo many Terms as there are Lives concerned. 2°. That the Probabilities of the other Lives furviving all the reft, are always expreffed each by one Term lefs than the preceding. 3°. That each firft Term of thofe whereby each Probability is ex- preffed, is always the Sum of all the other Terms ftanding above it. 4". That the Numbers 2, 6, 12, 20, 30, (ic. made ufe of in the Denominators of the Fraftions are generated by the Multiplication of the following Numbers, 1x2,2x3, 3x4, 4x5, &c. It would take up too much room to explain this general Rule in Words at length, for which Reafon I fhall content my felf with explaining only the Cafes of two and three Lives, which are the moft neceffary. And, Firft, if there be two Lives of a given Age, fuch as 40 and 70, take their Complements of Life, which as I have explained bfore, are the Differences between 86 and the refpeBive Ages, thofe Comple- ments therefore are 46 and 1 6. Divide the fiortefl Compkfnetit by the Double of the Lofigeji, and the ^otient will exprefs the Probability of the oldeft, Life Jwviving the youngcjl. Thus in the prefent Cafe, the fhorteft Complement being 16, and the double of the longeft being 92, I divide 16 by 92, and the Quo- tient -^ or-^ will exprefs the Probability required. Subtradl this Fradion from Unity, and the Remainder — will ex- ■' 2; prefs the Probability of the youngeft Life furviving the oldeft. So that the Odds of the youngeft Life furviving the oldeft, are 19 to 4. The Cafe of three Lives is thus : Suppofe there are three Lives of a given Age, fuch as 40, 45, and 60 ; take their refpedive Comple- ments of Life, which are 46, 41, 26, then divide the Square of the O o 2 ftiorteft: 2S4 1'he Doctrine of Chances applied fhorteft Complement by 3 times the Produd of the other two, and the Quotient will exprel's the Probability of the oldeft Life furviving the other two. Divide the middlemoft Complement by the Double of the greateft, and from the Quotient fubtraft the Square of the leaft divided by 6 Times the Produtt of the other two, and the Remainder will exprefs the Probability of the middlemoft Life furviving the other two. Subtraft the Sum of the two foregoing Probabilities from Unity, and the Remander will exprefs the Probability of the youngert Life furviving the other two. Thus in the Cafe propofed, the Probability of the oldefl: Life fur- (.76 viving the other two, will be found -pj = -jr nearly 56 ss The Probability of the middlemoft Life furviving the other two will be .131 ■=1 -^ nearly. The Probability of the youngeft Life furviving the other two will be — nearly, i PROBLEM XIX. Any Number of Lives being given^ to find the Probabi- lity of the Order of their Survivorjhip. Solution. Suppofe the three Lives to be thofe of A, B, C, and that it be re- quired to affign the Probability of Survivorfliip as limited to the Or- der in which they are written, fo that A fliall both furvive jB and C, and B alfo furvive C. This being fuppofed, let ft, p, q, reprefent the refpedive Complements of Life, of the youngeft, middlemoft, and oldeft, then the Probabilities of the fix different Orders that there arc in three things, will be as follows ; in Words thus j A 5, c A c. B B, A C B, c, A c. A B c, B, A 1 — 1 2f 1 __ J±_ J-JX- ' bnp __ ?? i«P bnf •y? tup Divide to the Valuation ^Annuities. 285 i". Divide the middlemoji Complement by the double of the greatefi, and let the §>uotient be fubtraEied from Utiity. 2". From that Remainder fubtraB again the ^lotient of the JJoorteJi Complement divided by the Double of tl.-- Middle nojl. 3°. To that new kemaind-r add the ^lotier.t arifmg from the Square ofthepoorteji Complement divided by fix times the PrcduSl of the great [jl and fiiiddlemofi ?nultiplied together, and this lajl Sum 'will exprefs the Probabiliiy of the firfl Order. The probability of the Second v/ill be found thus; » 1°. tiivide the Jlxrtefi Complement by the double of the middlemoji ^ and referve the ^otient. 2°. Divide the Square of the fiiorteft by three times the ProduCl of the longejl Complement^ multiplied by the Middlemofl^ and referve the new ^otient. 3°. Let the fecond ^wtient be fubtraSled from the firfl, and the Re- mainder -jiHl exprej} the Probability of the happening of the fecojid Order. The Probability of the third Order will be found as foUws. 1°. Divide the middlemoji Complement by the Double of the Greatefl, and referve the ^otient. 2°. Divide the jhortefi Complement by the Double oj the longefl, and referve the ^otient. 3°. Divide the Square of the Jhortefi Complement by fix times the Pro- duB of the longefi and middlemofi ?}iultiplied together, afid referve the ^otient. 4°. From the firfl ^otient referve d, fubtraB the fecond ; then to the Remainder add the Third, and the Refult of ihefe Operations tvill ex- prefs the Probability of the third Order. The Probability of the fourth Order will be found thus. 1°. Divide the fhortefi Complement by the Double oj the longefi, and referve the ^totient. 2°. Divide the Square of the fhortefi Complement by three Times the ProduB of the longefi and 7niddle-mofi, and referve the neaa Quotient. y. From the firfl ^otient refervcd, fubtraB the fecond, and the Re- mainder will exprefs the Probability of the fourth Order. The filth Order will be found as follows. Divide the Square of the Jhortefi Complement by fix times the ProduB of the longefi and middlemofi, multiplied together, and the ^otient ivill exprefs the Probability required. The Probability of the lafl Order is the fame as that of the fifth. PRO- 2 86 iChe Doctrine of Chances applied PROBLEM XX. D, whiljl in Healthy makes a Will, whereby he bequeaths 500 I. to E, and 300 1. to F. with this Condition^ that if either of them dies before him^ the whole is to go to the Survivor of the two ; what are the Values of the ExpeBations of ¥. and F, efiimated fro?n the time that the Will was writ P Solution. Suppofe D to be 70 Years of Age, E 36, and F 45 ; fappofc alfo that d reprefents the Intereft of i /. when Interefl is at 5 per Ce?it. An Annuity upon the Life of D is worth S-77y as appears from our Tables, which Value we may call M. Wherefore if it was fure that D would die before either of them, the Expedtation of E upon that Account, would b e worth in prefent Value 1 — dMy.^00, and the Expedlation of F, i — ^Mx300; which being reduced to Numbers, are refpedively t,S5 ^- ^5 ^' ^^^ ^ 1 3 /. 9 5. But as this depends on the Probability of D's dying firft, we are to look for that Probability, which is compofed of two Parts, that is, when the Order of Survivorfliip is either E, F, D, or F, E, D ; now the Order £, F, D, is the fame as A^ B, C, in the preceding Problem, whereof the Probability is i £ ^ + ~^ ' ^"^ the Order F, E, D, is the fame as B, A, C, whereof the Probability is -^^ — I — ^— , and the Sum of thofe Probabilities, viz. (,«1 I '' ► -i- -j- — ^ , will exprefs the Probability of D's dying before them both. Now the Ages being given, their Complements of Life will alfo be given, fo that n will be found = 50, p ■=/\.j, y = 16 ; for which reafon the Probability jurt; now fet down being exprefled in Numbers, will be 0.6865, and this being multiplied by the Expedations before found, viz. 315/. 15^. and 213/. g s. will produce 244 /. 35, 5^. and 146/. 10 s. %d and thefe Sums exprefs the prefent Expedations of E and F, arifing from the Profpedt of D's dying before either of them. But to the FahattoH (?/'Annuities. 287 But both £ and F have a farther Expedlation ; which, in refped to £, is, that he {hall furvivc Z), and that D fhall furvive F, in which Cafe he obtains 800 /. but this not being to be obtained before the Deceafe of Z), is reduced in prefent Value to 569/. 4 J- Now the Probability of obtaining this anfwers to the Order, A^ C, B, in the preceding Problem, which is exprefled by ^^ ^=0.1535; and therefore multiplying the Sum 569/. 4 j. by 0.1535, the Pro- du6t will be 87 /. j s. 5 d. and this will be the fecond Part of £'s Expedtation, which being joined with the firft Part found before, viz. 244 /. 3 s. ^d. the Sum will be 33 i /. 10 s. 10 d. which is the total Expectation of E, or the prefent Sum he might juftly expedt, if he would fell his Right to another. In the fame manner the total Expetflation of F will be found to be 213 /. 18 J. 6d. Otherwife, and more exadlly^ thus ; 1. Let the Value of an Annuity of 40 /. for D's Life, be taken off; which reduces the Sum to /, 569.2. as above. 2. The Heirs of D have likewife a demand upon this lafl Sum, for the Contingency of his outliving both the Legatees j which is implied tho' not exprefled in the Queftion. Subtradl therefore from the Value of the longeft of the 3 Lives D, E, F, which, by Prob. V, is 1 5.477, the Value of the longeft of the two Lives E, F; which, by Prob. IV, is 15.197; and the Remainder 0.28, D's Survivorship due to the Heirs, taken from /. 569.2, confidered as 20 Years Purchafe, or the Perpetuity, reduces it to /. 561.23. 3. This Sum, now clear of all demands, might be paid down immediately to £ and F, in the proportions of 5 and 3 , according to the Will; were their Ages equal. And altho' they are not, we fliall fuppofe that D, or his Executor named, pays it them in that manner j the Share of £ being /. 350.77, and that of F, /. 210.46. leaving them to adjuft their Pretenfions, on account of Age, between themfelves. 4. In order to which ; the Sums which £ and F have received being called G and L, refpeftively ; let the Value of £'s Survivorship after D and F, found as above, be denoted by e, and that of F after D and £ by/; Then thofe Values, e and /^ will reprefent the Chances, or Claims, which £ and Fhave upon each other's Sums L and G. And therefore the Ballance of their Claims ia ' '^~ ,-^ ;: due by For £ as the Sign ispofitive or negative. As. 288 He Doctrine of Chances applied As in our Example, ? and /being 3.26, 2.269 refpedivcly, E mufl: refund to F ( — ^T"^— = — ) /• 19.857 ; and the jull Values of their Legacies will be /. 330. 18 s. and /. 230. 6s. This laft Computation is to be ufcd when the Teftator D is not very old, or the Ages of E and Fare confiderably different ; or when both thele Conditions obtain: For in thofe Cafes, the Ratio of the Prcbabiiities of Survivorfhip will differ fenfibly from that of the Va- lues of the Probabilities reckoned ia Years purchafe. And the like caution is to be obferved in all fimilar Cafes. Of the ExpeSlations oj Life. I call that the Expedlation of Life, the Time which a Perfon of a given Age mayjuftly expedl to continue in being. I have found by a Calculation deduced from the Method of Fluxions, that upon Supposition of an equable Decrement of Life, the Expeda- tion of Life would be exprefTed by -«, fuppofing« to denote its Com- plement. However, if that Interval be once attained, there arifes a new Ex- pedation of -«, and afterwards of -^w, and fo on. This being laid down, I (hall proceed farther. PROBLEM XXL To find the Rxpe£iatio?i of two joint Lives^ that is, the 'Time 'which two Lives may expeSi to contimie together in beinz- Solution. Let the Complements of the Lives be « and p, whereof n be the longeft :ind/) the fhorteft, then the Expedtation of the two joint Lives, will ht-p ~ , in Words thus. From - thefhortefl Complement, fubtra5l the 6tb Part of its Square, divided by the greatef, the Remainder will exprefs the Number of Tears fought. Thus, fuppofing a Life of 40, and another of 50, the fhorteft Complement will be 36, the greateft 46, j of the ftiortefl will be 18, to the Valuation <?/*Annuities. 289 18, the Square of 36 is 1296, whereof the fixth Part is 216, which being divided by 46, the Quotient will be --!r- = 4.69 nearly; and this being fubtraded from 18, the Remainder 13.31 will exprefs the Number of Years due to the two joint Lives, Corollary. If the two Lives be equal, the Expecflation of the two joint Lives will be — part of their common Complement. PROBLEM XXIL Any Number of Lives being given^ whether equal or un- equal^ to find how many Years they may be expeSied to continue together. Solution. 1°. Take 4 of the fhorteft Complement. 2*. Take -^ part of the Square of the fhortefl:, which divide fuc- ceflively by all the other Complements, then add all the Quotients together. 3°. Take — part of the Cube of the fhorteft Complement, which divide fucceflively by the Producfl of all the other Comple- ments, taken two and two. • 4°. Then take — part of the Biquadratc of the fhorteft Comple- ment, which divide fuccefTively by the Produds of all the other Com- plements, taken three and three, and fo on. 5°. Then from the Refult of the firft Operation, fubtrad the Re- fult of the fecond, to the Remainder add the Refult of the third, from the Sum fubtrad the Refult of the fourth, and fo on. 6°. The laft Quantity remaining after thefe alternate Subtradions and Additions, will be the thing required. N' B. The Divifors 2, 6, 12, 20, &c. are the Produds of i by 2, of 2 by 3, of 3 by 4, of 4 by 5, (s'c. Corollary. If all the Lives be equal, add Unity to the Number of Lives, and divide their common Complement by that Number thus increafed by P p Unity, 200 T'he Doctrine of Chances applied Unity, and the Quotient will always exprefs the Time due to their joint Continuance. PROBLEM XXIII. Two Lives being given, to find the Number of Years due to the Longefi. SOLUTI ON. From the Sum of the Years due to each Life, fubtradt the Number of Years due to their Joint Continuance, the Remainder will be the Number of Years due to the Longeft, or Survivor of them both. Thus, fuppofing a Life of 40, and another of 50, the Number of Years due to the Life of 40, is 23 ; the Number of Years due to the Life of 50, is 18 ; from the Sum of 23 and 18, "viz. 41, fubtraft 13.3 1 due to their joint Continuance, the Remainder 27.69 will be the Time due to the longeft. Corollary. If the Lives be equal, then — of their common Complement will be the Number of Years due to the Survivor. Thus, fuppofing two Lives of 50, then their Complement will be 36; whereof two thirds will be 24; which is the Time due to the Survivor of the two. PROBLEM XXIV. Any Nujnber of Lives being given, to find the Number of Years due to the Longefi. S O L U T I ON. ' Let the Years due to each Life be refpedlively denoted by M, P, ^., 5, ^c. then let the joint Lives, taken two and two, be denoted by MP, iW^, M5, P^j &c. let alfo the j oint L ives, taken three and three be denoted by MP^, MFS^ M^, P^, See . More over, let the joint Lives, taken four and four, be denoted by MP^, &c. then if there be three Lives, the Time due to the longeft will be M— MP-fMP^ -VP— M§, +^-^^ ,. - But to the Valuation <?/'Annuities. 291 But if all the Lives be equal, let « be their common Complement, then the Time due to the longeft, will be -n. If there be four Li v es, the T ime due to the longed will be M— MP-{- MP^ -^MP^S ■^P—M^MPS 4-5 ^p^^p^s —PS But if all the Lives be equal, the Time due to the longeft will be exprefled by — of their common Complement. TJniverfally, if the common Complement of equal Lives be n, and the Number of Lives/*, the Number of Years due to the Longeft of them will be-^x«. PROBLEM XXV. Any Number of equal Lives being given, to find the 'Time in which one^ or two, or three, &c. of them will fail. SOLUT I ON. Let n be their common Complement, p the Number of all the Lives, q the Number of thofe which are to fail, then — f— x n will exprefs the Time required. In words thus j Multiply the common Complemefit of the Lives by the Number of the Lives that are to drop, and divide the ProduB by the Number of all the Lives increafed by Unity. Thus, fuppofing 100 Lives, each of 40 Years of Age, it will be found that 5 of them will drop in about two Years and a Quarter. But if we put t for the Time given, we fhall have the four follow- ing Equations; 1°. t-'^^ .0 2". qt=^' t±l P + lMt n ,0 A— V-i P p 2 In 4». n=^I±12lL 292 7he Doctrine of Chances applied In which any three of the four Quantities 71, p, q, t, being given, the fourth will be known. This Speculation might be carried to any Number of unequal Lives : but my Defign not being to perplex the Reader with too great Difficulties, I (hall forbear at prefent to profecute the thing any farther. PROBLEM XXVI. A, u^ho is 30 Tears of Age ^ buys an A?muity of i\. for a limited 1'ime of his Life^ f^PP^f^ * ^ Tears, on Con- dition that if he dies before the Expiration of that Time, the Purchafe Money is ivholly to be lofi to his Heirs ; to fnd the prefent Value of the Purchafe, fuppofing Interefl at 5 per Cent. Sol ution. Let n be the Complement of yf' s Life, m the limited Number of Years, p the Difference of n and m ; ^ the Value of an Annuity of I /. certain for m Years, and V the Value of the Perpetuity : then the t^'^elcnt Value of the Purchafe will be """^ ^'" ' . In Words 1 n thus ; 1°. Multiply V, the Value of a Perpetuity, at the given Rate of Intercjl, by m the limited Number of Tears, and referve the ProduSi. 2°. To the fame V add Unity, and take the Difference between their Sum and p, it'hich is the Excefs of the Complement of h's Age above the limited Number of Tears : multiply this Difference by Q, an Annuity certain for m Tears, to get the fecond ProduSi. 3°, Let the Sum of thefe Products, if p is greater than V-\- 1 ; and their Difference, if it is leffer, be divided by n, the Complement of Ks Age ; and the ^otient fhall be the Value of the Purchafe. As, in the Queftion propofed, where «:= 56, /«=:io, ^ = 46, ^-=7.7212, and V^=. 20 ; the f irft Produdt {mV) is that of 20 by 10, or 200. And p — V-\-\ being 46 — 21. -=25, the fecond Product is 25 X 7.72 1 2, that is 193.0302. The twoProduds added [p being greater than V-\-\) make 393.0302 : which divided by 56 quotes, for the Anfwcr, 7.0184 Years Purchafe. Note, I. When it happens that p is equal to V-\-i ; as, Interefl being at 5 per Cent, if the Difference of n and w is 2 1 ; the fecond Product ^ — V-\-i^ X ^vanifhing, the Anfwer is (imply — . 2. If: to the Valuation o/* Annuities. 293 2. \im^=.n^ or/>=:o, feeing V\% equal to —^, the Expreflion will be changed into — '■ ^^= ; which coincides with the So- r — I n X r — i lution of Prob. I : ^ reprefenting now the fame Thing as P did in that Problem. 3. By this Propoiition, fome ufeful Queftions concerning 7/7/j/r<7wa may be refolved. Suppofe A, at 30 Years of Age, affigns over to B an Annuity of 1000/ a Year, limited to 10 Years, and depending likewife upon A'i Life : then, by the foregoing Solution, A ought to receive for it only 7018/. 8i. Intereft being at 5 per Cent. But if B wants that the Annuity fliould ftand clear of all Rifques, he muft pay for i: the Value certain^ which is 772 i/. 45. and A ought to have his Life in- fured for 702 /. 1 6 j. the juft Price of fuch an Infurance being the Dif- ference of fhe Values of the Annuity certain, and of the fame Annuity fubjedt to the Contingency of the Annuitant's Life failing. The fame 702/. \bs. is likewife the Value of the Reverfion of this Annuity to a Perfon and his Heirs, who fliould fucceed to the Re- mainder of the 10 Years, upon ^'s Deceafe. See Prob. XXVIIL It is evident by the foregoing Procefs, that altho' the Queftion there propofed is particular, yet the Solution is general ; which Me- thod, often pradifed in my Dodlrine of Chances^ is of lingular Ufe to fix the Reader's Imagination. PROBLEM XXVir. A pays an Annuity of 1 00 1. durt7ig the hives of B and C, each 3 4 Tears of Age ; to find what A ought to give in prefent Money to buy off the Life of B, fup- pofing Inter efi at 4 per Cent. Solution. It will be found by our Tables that an Annuity upon a Life of 34 is worth 14,12 Years Purchafe ; and, by the Rules before delivered, that an Annuity upon the longeft of the two Lives of B and C is worth 18.40: hence it is very plain, that, to buy off the Life of 5, ^muft pay the Difference between 18 40 and 14.12, which being 4.28, it follows that yf ought to pay 428/. In the fame manner, if yf were to pay an Annuity during the three Lives of 5, C, D, whether of the fame or different Ages, it would be^ 294 ^^' Doctrine oJ Chances applied be eafy to determine what A ought to pay to buy off one of the Lives of jB, C, Z), or any two of them, or to redeem the whole. For, 1°. if the Life of £> is to be bought off, then from the Value of the three Lives, fubtrad the Value of the two Lives of fi and C, and the Remainder is what is to be given to buy off the Life of D. 2°. If the two Lives of C and D were to be bought off, then from the Value of the three Lives, fubtradl the Life of B, and the Remain- der is what is to be given to buy off thofe two Lives. Lajily, It is plain that to redeem the whole, the Value of the three Lives ought to be paid. PROBLEM XXVIIL A, wJoofe Life is worth 14 Tears Pur chafe., fuppofeiig In- ter efl at 4 per Cent, is to enjoy an Annuity, of 100 I. during the Term of '7^1 Years ; B and his Heirs have the Reverjion of it after the Deceafe of A for the Term remaining ; to f?id the Value of^ s Expeciation. Solution. Since the Life oi A'\s fuppofed to be worth 14 Years Purchafe when Intereft is at 4 per Cent, it follows from the Tables -that A muft be about 35 Years of Age, therefore find, by the twenty-fixth Propofition, the Value of an Annuity on a Life of 35, to continue the limited Time of 31 Years ; let that Value be fubtraded from the Value of an Annuity certain, to continue 3 i Years ; and the Re- mainder will be the Value of the Reverfion. PROBLEM XXIX. A is to have a?i Annuity of 100 \' for him and his Heirs after the failing of any o?ie of the Lives M, P, Q, the frfl of which is worth i 3 Tears Purchafe., the fecond 1 4, and the third i S , to find the prefent Value of his ExpeBation, Intereji of Money beijig fuppofed at 4 per Cent. Solution. By the Example to Prob. III. it appears, that an Annuity upon the above 3 joint Lives is worth 7,4 1 Years Purchafe ; let this be fup- pofed to the Valuation of A'snu it ies. 295 pofed =r R, and let / reprefent the prefent Value of a Perpetuity of 100/. which in this Caf e is 2500/. then the prefent Expectation of A will be worth i —dR xf. In Words thus; Multiply the Value of the three joint Lives by the hit ere jl of \\. then fubtraSfing that ProduB from Unity ^ let the Remainder be multiplied by the Value of the Perpetuity^ and the ProduB will be the ExpeBation required. In this Cafe 7.41, multiplied by 0.04, produces 0.2964, and this Produdl fubtraded from Unity, leaves 0.7036 ; now this Remainder being multiplied by 2500, produces 1759/. the Expedlation ol A. But if the Problem had been, that A fhould not have the Annui- ty before the Failing of any two of thofe Lives; from the Sum of all the joint Lives combined two and two, fubtradl the double Value of the three joint Lives, and let the Rem ainder be called T", then the Expedlation of A will be worth i — d'fxf; now, by the Rules be- fore delivered, we fhall find that the Sum of all the joint Lives com- bined two and two, is 29.06, from which fubtrading the double of the three joint Lives, 'y/2;. 1 4.82, th e P>.emainder is 14.24. Hence fup- pofing T'= 14.24, then 1 — ^Tx/will be found to be 1076/. and this is the Value ofA's Expedlation. Lajlly, If A was not to have the Annuity before the Extincftion of the three Lives, fuppofe the Value of the three Lives =z V, then the Expedlation oi A would be worth i—dVy.f, which in this Cafe is 465/. PROBLEM XXX. To determine the Fines to he paid for renewi7ig a?iy Num- ber of Years in a College-Leaf e of twenty ; afjd alfo what Rate of Interejl is made by a Purchafer^ who may happen to give an advanced Price for the fame, upon Suppoftion that the Contra&or is allowed 8 per Cent, of his Money. Altho' the Problem here propofed does not feem to relate to the Subjedl of this Book, yet as fome ufetul Conclufions may be derived from the Solution of it, I have thought fit to infert it in this Place. Table 2()6 The Doctrine of Chances applied Table of Fines. I 0.2146 8 2.2821 15 2 0.4463 9 2.6792 16 3 0.6965 10 3.1081 17 4 0.9666 11 3-57^3 18 5 1.2587 12 4'07i5 19 6 1-4133 13 4.6118 20 7 1.9144 14 5-^953 5-8254 6.5060 7.241 1 8.0349 8.9922 9.8I8I If a Purchafer gives the Original Contraftor r i Years Purchafe for his Leafe of 20, he makes above 6 ;- per Cent, of his Money. If he gives 1 2 Years Purchafe for the fame, he makes above 5 /. 8 j. per Cent, of his Money. If lie gives 13 Years Purchafe, he makes 4^ per Cent, of his Money. PROBLEM XXXI. To determine the Fines to be paid for renewing any Num- ber of Tears in a Col lege- Leafe of One and Twenty ; as alfo what Rate of Interefi is made by a Purchafer who may happen to give an advanced Price for thefame^ upon Suppofition that the ContraSlor is allowed 8 per Cent, of his Money. Table of Fines. 0.1987 0-4133 0.6450 0.8952 11653 1-4574 1. 6120 812,1131 9 2.2808 10 2.8779 ii;3.3o68 12 2>-77'^° 13 4.2702 14I4.8105 15 16 ^7 18 19 20 21 5-3940 6.0241 6.7047 7-439^ 8.2336 9.0909 10.0168 He that gives 1 1 Years Purchafe, inftead of 10.0168 for renevi'ing his Leafe for 21 Years, makes 61. 16 s. per Cent, of his Money. He who gives 12 Years Purchafe for the fame, makes very near 5 /. 16 s. per Cent, of his Money. He who gives 1 3 Years Purchafe for the fame, makes a little more than 4/. 16 s. per Cent, of his Money. The to the Valuation o/" Annuities. 297 The Values of Annuities for Lives having been calculated, in this Book, upon a Jiippofition that the Payments are made Yearly, and there being Jbme Occajions wherein it is /lipulated that the Payments Jhould be made Half-Tearly, I have thought Jit to add the tiaoj olio wing Problems; ichereby, i". // isJJxwn what the Half-Yearly Payments ought to be, if the Price of the Pur chafe is preferved. 2". Hoiv the Price of the Pur chafe ought to be increafed, if the Half-Tearly Payments are required to be the Half of the Tearly Payments. PROBLEM XXXII. An Annuity being given ^ to find what Half-Tearly Pay- ments will be equivale?it to it^ when Inter ejl of Money is 4, ^y or 6 per Cent Solution. Take Half of the Annuity, and from that Half fubtradl its 1 00th, or 80th, or 68th Part, according as the Iiitereft is 4, 5, or 6 per Cent. and the Remainder will be the Value of the Half- Yearly Payments required; thus, if the Annuity was 100/. the Half- Yearly Payments would refpedively be 49/. 10 s. 49/. js. 6d. ^.gl. ^s. 3^. nearly. PROBLEM XXXIII. 7^e prefent Value of an Annuity being given, to find how much this prefent Value ought to be increafed, when it is required that the Payments fljall be Half-Tearly, and alfo one Half of the Tearly Payments^ when Interefi is at /\., 5, or 6 per Cent. Sol u T I ON. To the prefent Value of the Annuity add refpedively its 99th, 79th, or 67th, and the Sums will be the Values increafed. As there are fome Perfons ivho 7nay be defirous to fee a general Solu- tion of the two lajl Problems, I have thought fit to add what follows. In the firflof the two lafk Problems, let A be the Yearly Payments agreed on, and B the Half- Yearly Payments required, r the Yearly Rate of Intereft, then B = -737- x A. In the fecond, let M be the prefent Value of the Yearly Payments, P the prefent Value of thofe that are to be Half- Yearly, then P= V~' x M. Qjl Table 298 7^(? Doctrine of Chances applied Table I. ^he prefent Value of an annuity of one pound, for any Number of Tears not exceeding 100, Interejl at 3 per Cent. k: >< -< >< n -1 Value. n en 26 Value. n -1 Value. n 76 Value. I 0.9709 17.8768 25-9512 29.8076 2 1-9135 27 18.3270 52 26.1662 77 29.9103 3 2.8286 28 18.7641 53 26.3750 7« 30.0100 4 3.7170 29 19.1884 54 26.5777 79 30.1068 .? 4-5797 30 19.6004 55 26.7744 80 30.2008 6 5-4172 31 20.0004 56 26.9655 81 30.2920 7 6.2303 32 20.3887 S7 27.1509 82 30.3806 8 7.0197 33 20.7658 5« 27.3310 l^ 30.4666 9 7.7861 34 21.1318 59 27.5058 84 30-5501 10 8.5302 3 5 21.4872 60 zj.bj^b «5 30.6311 II 9.2526 36 21.8323 61 27.8404 86 30.7099 12 9.9540 37 22.1672 62 28.0003 ll 30.7863 n JO. 6350 3« 22.4925 63 28.1557 88 30.8605 14 1 1. 2961 39 22.8082 64 28.3065 89 30.9325 15 11.9379 40 41 23.1148 23.4124 65 66 28.4529 28.5950 90 91 31.0024 16 12.561 1 31.0703 17 13.1611 42 23.7014 67 28.7330 92 31.1362 18 13-7535 43 23.9819'^ 68 28.8670 93 31.2001 »9 14.3238 44 24-2543 69 28.9971 94 3l.«622 20 14.8775 45 46 24.5187 24.7754 70 71 29.1234 29.2460 95 96 31.3224 21 15-4150 31.3809 22 15-9369 47 25.0247 72 29.3651 97 31-4377 23 16.4436 48 25.2667 73 29.4.807 98 31.4928 24 169355 49 25.5017 74 29.5929 99 31-5463 25 17-4131 50 25.7298 7S 29.7018 100 31-5984 The Value tf the Perpetuity is 33-1- Yeart Purchafi. Table to the Valuation (^/'Annuities. 299 Table II. Iloe prefent Value of an Annuity of one Pounds to continue fo long as a Life of a given Age is in being, Intere/l being ejlimated at 3 per Cent. Age. Value Age. 26 Value Age. 51 Value Age. 76 Value I 15-05 n-s^ 12.26 4.05 2 16.62 27 17.33 52 12.00 77 3-63 3 17.83 28 ij.xb 53 11-73 78 3-21 4 18.46 29 16.98 54 11.46 79 2.78 5 18.90 19.33 30 31 16.80 56 II 18 10.90 80 87 2.34 6 16.62 1.89 7 19.60 32 16.44 S7 10.61 82 1-43 8 19.74 33 16.25 58 10.32 83 096 9 19.87 34 16.06 59 10.03 84 0.49 10 19.87 36 15.86 60 61 9-73 9.42 8-; 0.00 11 J 9-74 15.67 0.00 12 19.60 37 15.46 62 9.11 ^3 19.47 38 15.26 ^3 8.79 14 19-33 39 ^S-<^S 64 8.46 »s 19.19 40 41 14.84 65 66 8.13 16 19.05 14.63 7-79 17 18.90 42 14.41 67 7-45 18 18.76 43 14.19 68 7.10 19 18.61 44 13.96 69 6.75 20 18.46 4? 46 1373 J 349 70 71 6.38 21 18.30 6.01 22 18.15 47 13-25 72 5-63 23 17.99 48 13.01 73 5-25 24 17-83 49 12.76 74 4.85 25 17.66 S<^ 12,51 7i 4-45 Qqa Table 3CO The Doctrine uf Chances applied Table III. He prefent Value of an Annuity of one Pound, for any Number of Yean not exceeding i oo, Intcrefi at y^ per Cent. ►< < H<; < n ?5 Value. n •-t Value. n 51 Value. n in 76 Value. J 0.9662 16.8904 23.6286 26.4799 2 1.8997 27 17.2854 52 237958 11 26.5506 3 2.8016 28 17.6670 53 239573 78 26,6190 4 3-6731 29 18.0358 54 24.1133 79 26.6850 5 45151 30 18.3920 55 24.2641 Ho 26.7488 6 5.3286 31 18.7363 56 24.4097 81 26.8104 7 6.1145 32 19.0689 57 24.5504 82 26.8700 8 6.8740 33 19.3902 58 24.6864 83 26.9275 9 7.6077 34 19,7007 59 24.8178 84 26.9831 lO 8.3166 35 20.0007 60 24.9447 85 27.0368 J X 9.0015 36 20.2905 61 25.0674 86 27.0887 12 9-6633 37 20.5705 62 25.1859 87 27.1388 13 • 10.3027 38 20.8411 63 25.3004 88 27.1873 M 10.9205 39 21.1025 64 25.41 10 89 27.2341 15 11-5174 40 41 21-3551 21.5991 65 66 25-5178 25.621 1 90 9' 27.2793 16 12.0941 27.3230 17 12.6513 42 21.8349 67 25.7209 92 27-3652 18 13.1897 43 22.0627 68 25-8^73 93 27.4060 19 13.7098 44 22.2828 69 25.9104 94 27.4454 20 14.2124 45 46 22.4955 22.7009 70 71 26.0004 26.0873 95 96 27-4835 21 14.6980 27.5203 22 15.^671 47 22.8994 72 26.1713 97 '^l-SSS'^ 23 15.6204 48 23.0912 73 26.2525 98 27.5902 24 16.0584 49 23.2766 74 26,3309 99 27.6234 25 16.4815 50 23-4556 7S 26.4067 100 27-6554 Thi Valut of the Perpetu ity is 284 Years Pur chafe. Table to the Valuatio?i of At^ f^ u it ies. 301 Table IV. ^he prefent Value of an Annuity of one Pound, fo long as a Life of a given Age is in being. Inter eft being efi mated ai ^^ per Cent. Age. Value Age. 26 Value Age. 51 Value Age. 76 Value 1 14.16 16. 2« 11.69 3.98 2 '5-53 27 16.13 52 11.45 77 3-57 3 16.56 28 15.98 53 11.20 7« 3.16 4 17.09 29 ^5-^3 54 10.95 79 2.74 5 17.46 30 31 15.68 5f 5^ 10.69 80 81 2.31 6 17.82 15-53 10.44 1.87 7 18.0^ 32 ^5-37 57 10.18 82 1.42 8 18.16 33 15.21 58 9.91 l^ 095 9 18.27 34 15-05 59 9.64 84 0.48 10 18.27 35 36 14.89 60 61 9-36 8c 861 0.00 11 18.16 14.71 9.08 0.00 12 18.05 37 14.52 62 8.79 13 17.94 38 H-34 63 8.49 14 17.82 39 14.16 64 8.19 ii 17.71 40 4' 13.98 65 66 7.88 7.56 16 17-59 '3-79 'I 17.46 42 ^3-59 67 7.24 18 J7-33 43 13.40 68 6.91 19 17.21 44 13.20 69 6.57 20 17.09 16.96 45 46 12.99 70 71 622 21 12.78 5-^7 22 16.83 47 12.57 72 5-5^ 23 16.69 48 12.36 73 5-H 24 16.56 49 12.14 74 4-77 25 16.42 50 11.92 75 4.38 TabLI: 302 Hoe Doctrine of Chances applied Table V. The frej'ent Value of an Anmdty of one Pound, for any Number of Tears not exceeding i oo, Intereft at 4 per Cent. I 2 3 4 J, 6 7 8 9 10 J I 12 13 25 16 17 18 »9 20 21 22 23 24 25 ►< Value. n to -1 « 0.9615 26 1.8860 27 2.7750 28 3.6298 29 4.4518 30 5.2421 31 6.0020 32 6.7327 33 74353 34 8.1108 35 36 8.7604 9.3850 37 9.9856 38 10.5631 39 1 1. 1 183 40 11.6522 41 12.1656 42 12.6592 43 13-1339 44 13-5903 45 46 14.0291 144511 47 14.8568 48 15.2469 49 15.6220 50 ^ Value. n CO 15.9827 16.3295 52 16.6630 53 169837 54 17.2920 55 56 17.5^^4 17-8735 57 18.1476 58 18.411 1 59 18.6646 60 57 18.9082 19.1425 62 J9.3678 63 19.5844 64 19.7927 ^5 66 19.9930 20.1856 67 20.3707 68 20.5488 69 20.7200 70 7J 20.8846 21.0429 72 21.1951 72 21.3414 74 21482 1 75 Value. ^^■7^75 21.8726 21.9929 22.1086 22.2198 22.3267 22.4295 22.5284 22.6234 22.7148 22.8027 22.8872 22.9685 23.0466 23.1218 23.1940 23.2635 23.3302 2 3 -3945 23.4562 23-5^5^ 23-5727 23.6276 23.6804 n Pi 70 77 78 79 I2 "87 82 83 84 85 l7 88 89 22. 91 92 93 94 95 96 97 98 99 100 Value. 2373 'I 237799 23.8268 23.8720 23-9153 23-9571 23.9972 24.0357 24.0728 24.1085 24.1428 24-1757 24.2074 24.2379 24.26 72 24-2954 24.3225 24.3486 24-3736 24-3977 24.4209 24.4431 24.4646 24.4851 24.5049 Table to the Valuation ^Annuities. 303 Table VL the prefenf Value of an Annuity of one Pound y to continue fo long as a Life of a given Age is in being. Inter ejl being efiimated <2/ 4 per Cent. Age. Value Age. Value Age. 1 Value Age. Value 1 13-36 26 15.19 51 11.13 76 3-91 2 H-54 27 15.06 52 10.92 77 3-52 3 15-43 28 14.94 53 10.70 78 3. II 4 ,5.89 29 14.81 54 IC.47 79 2.70 5 16.21 16.50 31 14.68 H-54 56 10.24 80 81 2.28 6 10.01 1.85 7 16.64 32 14.41 S7 ^-IJ 82 1.40 8 16.79 33 14.27 58 9-52 83 095 9 16.88 34 14.12 59 9.27 84 0.48 10 16.88 16.79 35 36 13.98 60 61 9.01 8.75 85 86 0.00 11 13.82 0.00 12 16.64 37 n-^7 62 8.48 13 i6.6o 38 1352 63 8.20 14 16.50 39 13-36 64 7.92 15 16.41 42 13.20 65 7.63 16 16.31 41 13.02 60 7-33 17 16.21 42 12.85 67 7.02 18 16.10 43 12.68 68 6.71 19 ^5-99 44 12.50 69 6-39 20 15.89 15.78 45 46 12.32 12.13 70 6.06 21 71 5-72 22 15.67 47 11.94 72 5.38 23 ^S'^S 48 11.74 73 5.02 24 ^5-43 49 11-54 74 4.66 25 »5-3» 50 11-34 IS 4.29 Tabli 304 7^^ Doctrine of Chances applied Table VII. The prefent Value of an Annuity of one Pound, for any vumber of Tears not exceeding 100, Intereft at 5 per Cent, >^. "< ^ " -< •1 Value. n -1 26 Value. n Value. 76 Value. J 0.9523 H-3751 18.3389 19.5094 2 1-8594 27 14.6430 52 18.4180 7Z 19.5328 '\ 2.7232 28 14.8981 53 18.4934 78 19-5550 4 3-5459 29 15.1410 54 18.5651 79 19.5762 <; 4-3294 5-0756 ■12 31 M-3724 55 56 18.6334 80 81 19.5964 6 15.5928 18.6985 19.6156 7 5.7863 32 15.8026 57 18.7605 82 19-6339 8 6.4632 33 16.0025 58 18.8195 «3 19.6514 9 7.1078 34 16.1929 59 18.8757 84 19.6680 10 7.7212 8.3064 35 36 16.3741 60 61 18.9292 «5 86 19.6838 Ti 16.5468 18.9802 19.6988 12 8.8632 37 16.71 12 62 19.0288 «7 19.7132 13 9-3935 3« 16.8678 63 19.0750 88 19.7268 14 9.8986 39 17.0170 64 1 9. 1 191 89 19.7398 M 10.3796 10.8377 40 41 17.1590 65 66 19.1610 90 91 19.7522 16 17-2943 19.2010 19.7640 ^7 11.2740 42 17.4232 67 19.2390 92 19.7752 18 1 1.6895 43 17-5459 68 19-2753 93 19.7859 IQ 12.0853 44 17.6627 69 19.3098 94 19.7961 20 12. 4622 12.8211 45 46 17.7740 70 71 19.3426 95 96 19.8058 21 17.8800 19-3739 19.8151 22 13.1630 47 17.9810 72 194037 97 19.8239 23 13.4885 48 18.0771 73 19.4321 98 19.8323 24 13.7986 49 18.1687 74 19.4592 99 19.8403 25 14.0939 5° 18.2559 75 19.4849 1 100 19.8479 Table io the Valuation of Annuities. 305 Table VIII. lie prefeiit Value of an Annuity of one Pound, to continue fo long as a Life of a given Age is in being, Interefl at j per Cent. Value Age. Value Age 11.96 12.88 26 ^3-37 27 13.28 28 13.18 29 13.09 30 12.99 ^I i2.8a 32 12.78 33 12.67 34 12.56 35 12.45 36 ^2-33 37 12.21 38 12.09 39 11.96 40 11.83 41 11.70 42 11.57 43 1^43 44 11.29 4? 1 1. 14 46 10.99 47 10.84 48 10.68 49 10.51 5° 10-35 R r Tablb 3o6 The Doctrine of Chances applied Table IX. ^he prefent Value of an Anmiity of one Pound, for any Number of Tean not exceeding loo, Inter eft at 6 per Cent. n I 2 3 4 _5 6 7 8 9 lO 1 1 12 13 H ii 16 ^7 18 19 20 21 22 23 24J 25 Value. 0-9433 '•^333 2.6730 3-4651 4.2123 4-9«73 5-5823 6.2097 6.8016 7.3600 7.8868 8.3838 8.8526 9.2949 9.7122 0.1058 0.4772 0.8276 1.1581 1.4699 1.7640 2.0415 2-3033 2-5503 27833 ^" -< ^ ft CD Value. so m 51 Value. 70 Value.. 26 13.0031 15.8130 16.4677 27 13.2105 52 15.8613 77 16.4790 28 13.4061 53 15.9069 78 16.4896 29 135907 54 M9499 79 16.4996 30 i3.764>^ 55 56 15.9905 80 81 16.5091 31 13.9290 16.0288 16.5180 32 14.0840 S7 16.0649 82 16.5264 33 14.2302 58 16.0989 83 16.5343 34 14.3681 59 16.1311 84 16.5418 35 14.4982 60 61 16.1614 85 86 16.5489 36 14.6209 16.1900 16.5556 37 H-7367 62 16.2170 87 16.5618 38 14.8460 63 16.2424 88 16.5678 39 14.9490 64 16.2664 89 16.5734 4.0 15.0462 65 66 16.2891 90 9^ 16.5786 41 15.1380 16.3104 16.5836 42 15.2245 67 16.3306 92 16.5883 43 15.3061 68 16.3496 93 16.5928 44 15-3831 69 16.3676 94 16.5969 45 15-4558 70 71 16.3845 95 96 16.6009 ^ 15-5243 16.4005 16.6046 47 15.5890 72 16.4155 97 16.6081 48 15.6500 73 16.4297 98 16.6114 49 ^5-7°75 74 16.4431 99 16.614^ 50 15.7618 7S 16.4558 100 16.6175 Tablb to the FaluatioH of Ahhvities. 307 Table X. The prefent Value of an Annuity of one Pound, to continue fo long as a Life of a given Age is in beings Intereji being ejlimated at 6 per Cent. Age. Value Age. 26 Value' 11.90 Age. 51 Value Age. 76 Value I 10.80 9-34 3.66 2 11-53 27 11.83 52 9.20 77 3-31 3 12.04 28 XI. 76 53 9.04 78 2.95 4 12.30 29 11.68 54 8.90 V^ 2.57 5 12.47 12.63 12 31 II. 61 i'-53 SI 56 8.72 80 81 2.18 6 8.56 1.78 7 12.74 32 II-45 57 8.38 82 1.36 8 12.79 33 11.36 58 8.20 l^ 0.92 9 12.84 34 11.60 S9 8.C2 84 0.77 10 12.84 12.79 35 36 11.18 60 61 7-83 85 86 0.00 1 1 11.09 763 coo 12 12.74 2,7 1 1. 00 62 7.42 13 12.69 38 10.90 63 7.21 H 12.63 39 10.80 64 7.00 15 12.58 40 41 10.70 10.60 65 6.77 16 12.53 66 6.53 17 12.47 42 10.50 67 6.22 18 12.41 43 ^o-37 68 6.03 ^9 12.36 44 10.26 69 S-77 20 12.30 ia.23 4T 46 10.14 70 71 5-50 5.22 2? 10.02 22 12.17 47 9.90 72 4-93 23 12. II 48 9.76 73 4-63 24 12.04 49 9-63 74 4-32 25 11.97 50 9.49 7S 4.00 R r 2 Note } 3o8 Ihe Doctrine of Chances applied Note ; The i/?, 3^, 5//', jth and gth Tables ferve like wife to refolve the Queftions concerning Compound Interejl : as I. To find the prefent Value of loool. payable 7 Tears hence, <?/ 3r per Cent. From the prefent Value of an Annuity of i / certain for 7 Years, which, in Tab. III. is 6. 1 145, I fubtradt the like Value for 6 Years, which is 5.3286; and the Remainder 7859 is the Value of the ']th Year's Rent, or of i /. payable after 7 Years ; which mul- tiplied by 1000 gives the Anfwer 785 /. \%Jh. ir. If it is alked, ischat will be the Amount of the Sum S in 7 Tears at i^ per Cent ? Having found .7859 as above, 'tis plain the Amount will be 7*S9 III. If the Queflion is. In ivhat time a Sum S iti/l be doubled, tripled, cr increafed in any given Ratio at 3, 3I, &c. per Cent. I take, in the proper Table, two contiguous Numbers whofe Difference is neareft the Reciprocal of the i?tf//(j given, as J, j, &c. And the Year againft the higher number is the Anfwer. Thus in Tab. I. againft the Years 22, 23, ftand the Numbers IC.9369 and 16.4436; whofe Difference .5067 being a little more than .5, or — , fliews that in 23 Years, a Sum aS will be a little lefs than doubled, at 3 per Cent. Compound Intereft. And againft the Years 36 and 37 are 21.8323, and 22.1672 ; the Difference whereof being .3349> nearly ~ , ftiews that in 14 Years more it will be al- moft tripled. If more exaftnefs is required ; take the adjoining Difference whofe Error is contrary to that of the Difference found ; and thence com- pute the proportional part to be added or fubtrafted thus, in the laft of thefe Examples, the Difference between the Years 37 and 38 is .3252, which wants .0081 of .3333 (=7) . as the other Difference .3349 exceeded it by .0016. The 38^/6 Year is therefore to be di- vided in the Ratio of 16 to 81 ; that is -^-^ of a Year, or about 2 Months, is to be added to the 37 Years. IV. To. to the Valuation 0/ AimuiTits. 309 IV. To find at ivhat Rate of Inter eft I ought to lay out a Sum S, fo as it may encreafe ^ for Inftance, or become — S /;; 7 Tears. Here th& Fradion I am to look for among the Differences is — , or the De- 4 cimal .j^ ; which is not to be found in Tab. I. or III, till after the limited Time of 7 Years. But in Tab. V, the Numbers againft 6 and 7 Years give the Difference .7599 ; and the Rate is 4 per Cent. nearly. To find how nearly ; we may proceed as under the foregoing Rule. Take the Difference between 6 and 7 Years in Tab. VII. for f/^r Cent. ; which being -7107, wanting .0393 of .75, as 7599 exceeded it by .0099 ; divide Unity in the Ratio of 99 to 393, that is of 33 to 131, and the leffer Part added to 4 per Cent, gives the Rate fought,, 1.7, I 1- 104' ''•5 PART PART II. Contahiing the Demonjlrat'ions of fome of the principal fropopitions i?i the foregoing Treatife, CHAPTER I. IOhferved formerly, that upon Suppofition that the Decrements of Life were in Arithmetic Progreffion, the Conclufions derived from thence would very little vary from thofe, that could be de- duced from the Table of Obfervations made at Brejlaw, concerning the Mortality of Mankind ; which Table was about fifty Years ago inferted by Dr. Halley in the Philofophical TranfaSlions, together with fome Calculations concerning the Values of Lives according to a given Age. Upon the foregoing Principle, I fuppofed that if n reprefented the Complement of Life, the Probabilities of living i, 2, 3, 4, 5, (ifc. Years, would be exprefled by the following Series, ^-^ , ^^ , ^^ , '■^ , ^^ , (Sc. and confequently that the Value of a Life, whofe Complement is «, would be exprefled by the Series ---' 4- "^^ 4- ^ 4- "^ -\- ^ , (j^c. the Sum of which I have .-- Ip aflerted in Problem L to be — , where the Signification of O' the Quantities P and r is explained. As the Reafonings that led me to that general Exprefllon, require fomething more than an ordinary Skill in the Doftrine of Series, I fhall forbear to mention them in this Place j and content myfelf with pointing out to the Reader a Method, whereby he may fatisfy himfelf of the Truth of that Theorem, provided he underftand fo much of a Series, as to be able to fum up a Geometric Progreffion. Demon- to the Valuation of Annvirits. 311 Demonstration. Therefore, And ■:^=-L + -i--l--L-4-^-|— i-.. . . + -1-. Therefore, rt" n — I r I I I n n nr nrr nr^ nr'' ' ' ' „^— i But this is to be divided by r — r, or multiplied by r— I r ^^ rr ~ ri ~ r* i^ rS ^^ rb > ^'■• Then multiplying adluaUy thofe two Series's together, the Produ<ft will be found to be :Lz1 1 J : ! L.Af^. nr m r nri nr* ni S nr* ' 7I> ^&C. T^ nri nr* n.s »r« ^'" «,♦ ~~ «,J ,r° ^'^^ And adding the Terms of the perpendicular Columns together wc- fhall have ^+-^^ + ^+^-1- J^L . J:z± ^^ nr ' itrr • nr> • ni* • nri I aii *-'•'• which confequently is equal to ^_\ : which was to be demon- ll rated. If it be required that upon the Failing of a Life, fuch Part of the Annuity fiiould be paid, as maybe proportional to the Time elapfed from the Beginning of the laft Year, to the Time of the Life's failing, then the Value of the Life will be ;^, —^P, wherein a reprefents the hyperbohc Logarithm of the Rate of Intereft. But- 312 n^e Doctrine (?/" Chances applied But becaufe there are no Tables printed of hyperbolic Logarithms, and that the Redudion of a common Logarithm to an liyperbolic is fomevvhat laborious, it will be fufficient here to fet down the hyper- bolic Logarithms of 1.03, 1-035, ^•°4> ^•°5> i-o6, which are re- fpedively, 0.02956, 0.0344, 0.03922, 0.04879, 0.05825, or — , — , — , — , — nearly. 35 ' 3' 5' 4' '°3 ^ CHAPTER n. "Explaining the Rules of combined Lives. Siippofing a fiditious Life, whofe Number of Chances to continue in being from Year to Year, are conftantly equal to a, and the Num- ber of Chances for foiling are conftantly equal to b, fo that the Odds of its continuing during the Space of any one Year, be to its failing in the fame Interval of Time conftantly as a to b, the Value of an Annuity upon fuch a Life would be eafily found. For, if we make a-Yb=s, the Probabilities of living i, 2, 3, 4, 5, ^e. Years would be reprefented by the Series -^, 11, 1- " ' J > /f > jS » &c. continued to Eternity -, and confequently the Value of an Annuity upon fuch Life would be exprefled by this new Series -1. _! — i^_| — ^ — I — 1^ &c. which being a geometric Progreffion perpetually decreafing, the Sum of it will be found to be -~^ : thus, if a ftands for 21, and b for i, and alfo r for 1.05, the Value of fuch Life would be ten Years Purchafe. From thefe Fremifes the following Corollaries may be drawn : Corollary I. An Annuity upon a fidlitious Life being given, the Probability of its continuing one Year in being is alfo given ; for let the Value be Corollary n. If a Life, whofe Value is deduced from our Tables is found to be worth 10 Years Purchafe, then fuch Life is equivalent to a fiditious Life, whofe Number of Chances for continuing one Year, is to the Number of Chances for its failing in that Year, as 21 to 1. C R L- to the Valuation of Ann ui ties. 313 Corollary III. Wherefore having taken the Value of a Life from our Tables, or calculated it according to the Rules prefcribed j we may transfer the Value of that Life to that of a fidlitious Life, and find the Num- ber of Chances it would have for continuing or failing Yearly. Corollary IV. And the Combination of two or more real Lives will be very near the fame as the Combination of fo many correfponding Jikitious Lives ; and therefore an Annuity granted upon one or more real Lives, is nearly of the fame Value as an Annuity upon a fiditious Life. Thefe things being premifed, it will not be difficult to deterrpine the Value of an Annuity upon two or three, or as many joint Lives as may be afligned. For let X reprefent the Probability of one Life's continuing from Year to Year, and y the Probability of another Life's continuing the fame Time ; then according to tlie Principles of the Dodrine of Chances, the Terms X)', xxyy, x"^ y^, x^ y'^, x^ y'-, Zjc. will refpeftively reprefent the Probabilities of continuing together, i , 2, 3, 4, 5, Gff. Years J and the Value of an Annuity upon the two joint Lives, will be -^ + -^ + -f- _}- _i- _|- ^ ^^. which being a Geometrical ProgrefTion perpetually decreafing, the Sum of it will be found to be -~- : let now M be put for the Va- lue of the firfl Life, and P for the Value of the fecond, then by our firft Corollary It appears that x = -~~ , and y=-~-- ; and there- fore having written thefe Values of x and y in the Expreflion ■ V ■ , which is the Value of the two joint Lives, it will be changed into a;+, y pjJ,-MPr ' "^^^^^ ^s ^^^ ^^^^ Theorem that I had given in my firft Edition. It is true that in the Solution of Prcl>. II. I have given a Theorem which feems very different from this j making the Value of the joint Lives to be -mZ^pztTmT • wherein d reprefents the In- tereft of i /. and yet I may affure the Reader, that this laft Expref- fion is originally derived from the firft ; and that whether one or the S f othe 314 '^^ Doctrine 0/" Chances applied other is ufed, the Conclufions will very little differ : but the firft Theorem is better adapted to Annuities paid in Money, it being cuftomary that the lafl: Payment, whether it be Yearly or Half- Yearly, is loft to the Purchafer; whereas the fecond Theorem is better fitted to Annuities paid by a Grant of Lands, whereby the Purchafer makes Intereft of his Money to the laft Moment of his Life : for which Reafon I have chofe to ufe the laft Expreflion in my Book. By following the fame Method of Inveftigation, we fliall find that if M, P, ^, denote three fingle Lives, an Annuity upon thofe joint Lives will be — .^— =i_-,2xl— — -— — , in the Cafeof Annuities paya- "ble in Money; or mi'^m^+pZ- 2 aM -■> ' ^" *^^ ^^^^ °^ Annuities paid by a Grant of Lands. CHAPTER III. Contaim?ig the Demonftration of the Rules given in Pro- blems ^th and e^thy for determining the Value of longefi Life, Let X and y reprefent the refpeftive Probabilities which two Lives have of continuing one Year in being, therefore i — a: is the Proba- bility of the firft Life's failing in one Year, and i — y the Probabi- lity of the fecond Life's failing in one Year: Therefore multiplying thefe two Probabilities together, the Produdl 1 — x — y-\-xy will re- prefent the Probability of the two Lives failing in one Year ; and if this be fubtrafted from Unity, the Remainder a; Ary — ^^y will exprefs the Probability of one at leaft of the two Lives outliving one Year : which is fufficient for eftabliftiing the firft Year's Rent. And, for the fame Reafon xx-\-yy — xxyy will exprefs the Probabi- lity of one at leaft of the two Lives outliving two Years : which is fuffi- cient to eftablifh the fecond Year's Rent. From the two Steps we have taken, it plainly appears that the longeft of two Lives is cxprefiible by the three following Series ; r T^ rr I r! I r+ ' fS I + f+f + 4 + ^+f^'!-&c. xy xxyy fciy'i x*y* jr'^S i r rr ri r* r' J Whereof to the Valuation p/'Annuities. 315 Whereof the firft reprefents an Annuity upon the firft: Life, the fe- cond an Annuity upon the fecond Life, and the third an Annuity up- on the two joint Lives ; and therefore we may conclude that an An- nuity upon the longeft of two Lives, is the Difference between the Sum of the Values of the fmgle Lives, and the Value of the joint Lives : which have been exprefled in Problem IV. by the Symbols M-![-P—MK In the fame manner it will be found that if x, y, 2, reprefent the refpedive Probabilities of three Lives continuing one Year, then the Probability of their not failing all three in one Year will be exprefled by x-\-y-\-z — xy — xz — yz-\-xyz ; which is fufficient to ground this Conclufion, that an Annuity upon the longeft of three Lives, is the Sum of the fingle Lives, minus the Sum of the joint Lives, plus the three joint Lives : which has been expreflTed by me, by the Symbols Af-f P-1-^— mP— M^— 7^-f m7^ From the foregoing Conclufions, it is eafily perceived how the Value of the longeft of any Number of Lives ought to be determined ; 'VIZ. by the Sum of the Values of the fingle Lives, mi/ius the Sum of the Values of all the joint Lives taken two and two, plus the Sum of all the joint Lives taken three and three, minus the Sum of all the joint Lives taken four and four, and fo on by alternate Additions and Subtra6lions. CHAPTER IV. Containing the Demonjiratiojts of what has been faid con- cer?nng Reverjtons, and the Value oj one Life after one or more Lives. 1°. It plainly appears that the prefent Value of a Reverfion after one Life, is the Difference between the Perpetuity, and the Value of the Life in Poffeflion : Thus, if the Life in Poffeffion he worth 14 Years Purchafe, and that I have the Reverfion after that Life, and have a mind to fell it, I muft have for it 1 1 Years purchafe, which is the Difference between the Perpetuity 25, and 14 the Value of the Life, when Money is rated at 4 pe?- Cent. 2°. It is evident that the Reverfion after two, three, or more Live?, is the Difference between the Perpetuity, and the longeft of all the Lives. S f 2 But 3^6 'llie iDocTRiNE 6/" Chances applied But the Value of a Life after one or more Lives not being fo ob- vious, 1 think it is proper to infill: upon it more largely : let x there- fore reprefcnt the Probability of the Expedant's Life continuing one Year in being, and_)' the Probability of the fecond Life's continuing alfo one Year in being, and therefore i — y is the Probability of that fe- cond Life's filling in that Year ; from which it follows, according to the DoiSrine of Chances, that the Probability of the firft Life's continuing one Year, and of the fecond's failing in that Year, is X X I — y, or X — -xy ; which is a fufficient foundation for drawing the following Conclufion, 'ciz. that the Value of the firft Life after the fecond is the Value of that firft Life minm the Valu e of t he two joint Lives : v/hich I have exprefled by the Symbols M — MP. In the fame manner, if x, y, z, reprefent the refped tiv e Pro babi- lities of three Lives continuing one Year, then x X i — y X i — z, Will reprefent the Probability of the firft Life's continuing one Year, and of the other two Lives failing in that Year ; but the foregoing Ex- preflion is brought, by adual Multiplication, to its Equivalent x — x y — xz-\-xyz ; from whence can be deduced by meer Infpedlion the Rule given in Prob. VIII. "oiz. that the prefent Value of the firft Life's Expedation after th e Failing of the other two, is M—MP—'M^YMP^ CHAPTER V. Containing the Demonjlration of what has been ajferted in the Solution of the lOth a?td 2(^th Problems. In the Solution of the loth Problem, M'" denoting the prefent Value of an Annuity to continue fo long as three Lives of the fame Age fubfift together, let us fuppofe that n denotes the Number of Years during which the Annuity will continue ; then fuppofing r to exprefs the Rate of Intereft, it is well known that the prefent Value of that I n Annuity will be — — — , wherefore we have the Equation M"*-=. — _'" , or making r--^ iz=d, M'" = — -^ , from whence will be deduced -^= i — </M"', and confcquently r" = ■ ^_J,^^-„ . Now let us fuppofe that a Sum / is to be received to eternity at the equal to the Valuation <?/ Annuities. 317 equal Intervals of Time, denoted by «, and that we want to find the prefent Value of it ; it is plain to thofe who have made fome Profi- ciency in Algebra, that —7— is the prefent Value of it, let us there- fore in the room of r" fubftitute its Value found before, viz. \_jyi ■■ » and then r' — i will be found equal to —^r > ^"^ confequently -J——l:dJ£L x/: as in the Solution of Frob. X. Now it will be eafy to find n ; for let us fuppofe ^^jj^^T'jthen r" r=: jT. and therefore n = -^ — . The 29th Problem has fome Affinity with the loth; in the former it was required to know the prefent Value of a Sum f, paya- ble at the Failing of any one of three equal Lives, but in the latter the three Lives are fuppofed unequal ; but befides, it is extended to two other Cafes, viz. to the prefent Value of a Sum / to be paid after the Failing of any two of the Lives, as alfo to the prefent Va- lue of a Sum /to be paid after the Failing of the three Lives. For in the firft Cafe, let us imagine an Annuity to be paid as long as the three Lives are in being ; or, which is the fame thing, till one of the Lives fails ; and let us fuppofe that R reprefents the Value ot the three joint Lives ; let us alfo fuppofe that n is the Number ot Years after which this will happen, and that d is the Intereft of i /. therefore -4" is the prefent Value of the Sum /to be then paid ; but I R= — 7-^, therefore — = i •— d R, and therefore — - := i—dR^f. But the fecond Cafe has fomething more of Difficulty, and there- fore I fhall enlarge a little more upon it : let us imagine now that there is an Annuity to continue not only as long as the three equal Lives are in being, but as long as any two of the faid Lives are ia being ; now in order to find the prefent Value of the faid Annuity, let us fuppofe that x, y, z, reprefent the refpedive Probabilities of the faid Lives continuing one Year. Therefore. 1". X y z reprefents the Probability of their all outliving the Year. 2°. xyy. I — z, or xy — xyz reprefents the Probability of the two firft outliving the Year, and of the third failing in that Year. 3i8 T*he Doctrine o/" Chances applied 3°. xz X I — y or xz — xyz reprefents the Probability of the firfl and third's outliving the Year, and of the fecond's failing in that Year. 4°. yxv^x — 2, or^z — xyz reprefents the Probability of the fe- cond and third's outliving the Year, and of the firft's failing in that Year. Then adding thofe feveral Produ6ts together, their Sum will be found equal to xy ■\-xz-\-yz — axyz, which is an Indication that the prefent Value of an Annuity to co ntinue as long as two of the faid Lives are in being is WP -\-M^-\-^^ — 2MP^, which we may fuppofe = T. Let us now compare this with an Annuity certain to continue n Years, the Rate of Intereft being fuppofed := r, and r — i = ^, I then we (hall have the Equation — - — = T, from whence we (hall find--^=i — dT, and confequently -^ , which is the prefent Va- lue of the Expedlation required, is =i — dTyf. By the fame Method of Procefs, we may find the prefent Value of an Annuity to continue fo long as any one of the three Lives in queftion is fubfifting ; for let x, y, z, reprefent the fame things as before. 1°. xyz reprefents the Probability of the three Lives outliving the firfl: Year. 2°. xy-\-xz-\-yz — ^'^y^ reprefents the Probability of two of them outliving the Year, and of the third's failing in that Year. 3°. XXI — JX 1 — z, or X — xy — xz-\-xyz reprefents the Pro- bability of the firft Life's outliving the Year, and of the other two failing in that Year. 4°. y X I — X X I — z, or y — xy — zy-\-xyz reprefents the Pro- bability of the fecond Life's outliving the Year, and of the other two failing in that Year. 5°. 2; X I —AT X I — Jj or z — xz — _yz-{-xy2r reprefents the Pro- bability of the third Life's outliving the Year, and of the other two failing in that Year. Now the Sum of all this is x-\-y-{-z — xy — xz — yz-\-xyZi which is an Indication that the Value of an Annuity to continue as lont^ as any one of three Lives is in being ought to be exprefTed by M-f P-{-^^j--MF—M^}—P^-\- MF^: and this laft Cafe may be looked upon as a Confirmation of the Rule given in our cth Problem. CHAP- to the Valuation <?/"Annuities. 319 CHAPTER VI. Containing the Demonjiration of "cohat has been faid con- cerning fuccejftve Lives in the Solution of Prob. XIII. What has been there faid amounts to this ; The prefent Values of Annuities certain for any particular Number of Years being given, to find the prefent Value of an Annuity to continue as long as the Sum of thofe Years. Let us fuppofe that M reprefents the prefent Value of an Annuity to continue n Years, and that P reprefents the prefent Value of an- Annuity to continue p Years ; the firft: Queftion is. how from thefe Data to find the prefent Value of an Annuity to continue « -{-/' Years, the Inveftigation of which is as follows : let r be the Rate of Intereft, and fuppofe r — i which denotes the Intereft of i /. -=1 d ; I n then, 1°. M=:r — -f— , therefore -^ = i — dM; and for the fame Reafon — = i — dF. Therefore "qi^ = 1 — d M % I — dPr= I — dM— dP -\- ddMP. Let now /be fuppofed to be the Value of the Annuity which is to continue n-\-p Years, then -~^ —i-~df. Therefore i — df= i —dM— dP + ddMP; then fubtrading Unity on both Sides, dividing all by d, and changing the Signs, we fliall have/= Af + P — dMP. 2°. By the fame Method of Procefs, it will be eafy to find that if My P, ^, reprefent Annuities to continue for the refpedlive Num- ber of Years n, p, q, then the Value of an Annuity to continue n+p-Vq Years will be M-\-P-\-Sl — dMP — dMSlj—dP^ •\- ddMP^: the Continuation of which is obvious. Let us now fuppofe that the Intervals «, p, q, are equal, then the Values M, P, ^, are alfo equal ; in which Cafe, the foregoing Ca- non will be changed into this, 3M — idMM-\-d- M-, or — -j : but if this Numerator be lubtracted from Unity, the Remainder will be i — 3J/V/-]-3iWMM — d^M^=z I — dM\' ; and fubtrading this again from Unity, the original Nu- merator will be reftored, and will be equivalent to i — i — d M ^y. and confequently, if M reprefents the Value of an Annuity to con- tinue 320 The Doctrine of Chances applied 7a?»! tinue a certain Number of Years, then ^ — ' -' ^ ■-■ ■ will reprefent the Value of an Annuity to continue three times as long. And univerfally, if M ftands for the Value of an Annuity to con- tinue a certain Number of Years, then '~'~'^ — will reprefent the Value of an Annuity to continue n times as lo ng. And if n were infinite, I fay that i — di\iy would be r= o ; from whence the Value would be = -^ or -^ , which reprefents the Value of the Perpetuity. But that there may remain no fcruple ab out what w e have afferted above, that in the Cafe of n being infinite, i — ^Ml" would vanifli j I prove it thus, ^^M^ therefore i b. ^M, therefore i — ^M is a Fradtion lefs than Unity : now it is well known that a Fradion lefs than Unity being raifed to an infinite Power, is nothing, and was therefore fafely negleded. CHAPTER VII. Co72tainifjg the Detnonjlration of what has been afferted in the 2'2-d and 3 3<a^ Problems concerning half yearly Pay- ments \ as alfo the Invefligation offo7ne Theorems relating to that SubjeB. It is well known that if an Annuity A is to continne « Years, the A-± prefent Value of it is ■ ^f ■ j fuppofing r to reprefent the Rate of Intereft ; now to make a proper Application of this Theorem to half-yearly Payments, I look upon n as reprefenting indifferently the Number of Payments and the Number of Years ; let us now fup- pofe a half-yearly Rent B of the fame prefent Value as the former, and 10 continue as long, then the Number of Payments in this Cale will be 2;?, but the Rate of Intereft, inftead of being r, is now rr, which being raifed to the Power 2», will be r" as before ; for which B-IL Rcafon the prefent Value of the half-yearly Payments is ^ : n —1 b Jt by Hypothefis, the prefent Values of the yearly and half-yearly Pay- to the Valuation ^Annuities. 321 A-± n-JL. n r Payments are the famej therefore • ,._[ = . '^ , and dividing n — f both fides of the Equation by i ^ , we (hall have = I from whence will be deduced 5 = ^[^' x yl : and in the fame manner, if the Payments were to be made quarterly, then B would I be = ^ ■ " ■ ' ■ X ^i and fo on. r — I But if we fuppofe that a Rent fhall be paid half-yeaily, and that it fhall be alfo one half of what would be given for an annual Rent, and that the two Rents {hall be of the fame Duration ; then the pre- fent Values of the yearly and half-yearly Rents will be different : for let Mand P be the prefent Values of the yearly and half-yearly A — — ' \A-i:^ Rents, then M=. '— , and P=—^ , and dividins; both Va- J J I lues by -<^ , we (liall have M, P :: , — '■ — ; and con- fequently P = ^ ^J_ x M. The two laft Problems bring to my Mind an Affertion which was maintained, about fix Years ago, in a Pamphlet then publiflied ; which was that it would be of great Advantage to a Perfon wjio pays an Annuity, to difcharge it by half-yearly Payments, each of one half the Annuity in Queftion : the Reafon of which was, that then the time of paying off the Principal would be confiderably (hortened. I had not the Curiofity to read the Author's Calculation, becaufe I thought it too long ; fince which Time I thought fit to examine the thing, and found that indeed the Time would be fhortened, but not fo confiderably as the Author imagined : which to prove, 1 fuppofed a Principal of 2000/. an Annuity of 100/. and the Rate of Interefi: 1.04: in confequence of which, I found that the Principal would be difcharged in 41 Ycarsj this being founded on the general Theorem A-J- n • ^_\ = P, in which A reprefents the Annuity, P the Principal, r the Rate of Interefl:, and « the Number of Years : now to apply this to the Cafe of half-yearly Payments, let us luppofe that p denotes the Number of Years in which the Principal will be difcharged ; there- fore 2 ^ will be the Number of Payments, -^^ the Annuity, and ;•• the Rate of Interefl : which being refpedively fubflituted in the Room of T t • ;;. 32 2, 'The Doctrine of Chances applied », A, r, we fhall have now - . ' -^zP, hutr- — 1 = 0.010804, ' rz — I which being fuppofed = «, we (hall have t-4— ^ ■— = m. P, and — z=z\A — mP, or -^=:_ 10. ■? 02} therefore -^ = — ^- — , or rP = ''° , and p log. rz=log. 50 — log. 10.392 =0.6822709, therefore/. = ^^-^-^ J again, log. r = 0.0170333, therefore /> =: 0.06822700 ■'■S-r - = 4.0.0;: and therefore the Advantage of paying half- 0017033} ^ ^ . . ^ ,^ V yearly would amount to no more than gaining one Year in 41. Quarterly Payments, or half- quarterly, nay even Payments made at every Inftant of Time, would not much accelerate the Difcharge oif the Principal. Which to prove, let us refume once more our ge- n neral Theorem ■ __'' =: P -, let us now imagine that the Number of Inftants in the Year is = /, let us further fuppofe that s is the Number of Years in which the Principal will be difcharged, then in the room of yf, writing y-^j in the room of r, writing r' ; and in the room of «, writing s t, we fhall have ^ =: P. But it is known, that if / reprefents an infinite Number, fuch as is the; I Number of Inftants in one Year, then r' — i = y /o^. r, we have, -j — L- ^ — r therefore '_ i__= -P. or -j^ = Pj let the Logarithm of r A A be fuppofed = a^ therefore A - = aP, and— =yf — aP^ and r' = -j^p , which fuppofe = ^, then s = -^-^ : But it is to be noted, that a reprefents the hyperbolic Logarithm of r, which,; is, as we have feen before, 0,0392207 whenr ftands for 1,04; this being fuppofed, the Logarithm of ^will be found to be 0,6663 794^., which being divided by the Logarithm of r viz. 0,0170333, the- Quotient to the Valuation o/'Annuities. 323 Quotient will be 39,1 Years; but in this laft Operation the Loga- rithms of ^ and r, may be taken out of a common Table. CHAPTER Vlir. Containing the Detnonjlration of what has been [aid con- cerning the Probabilities of Survivorpip, What I call Complement of Life having been defined before pag. ibi^. I fhall proceed to make ufe of that Word as often as occafion fhall require. Hypothesis. A B C D E F G S Let it be fuppofed that the Complement of Life A S being divided into an infinite Number of equal Parts reprefenting Moments, .he Probabilities of living from A to B, from A to C, from A to D, &c. are refpedlively proportional to the feveral Complements SB, S C, S D, in fo much that thefe Probabilities may relpedively be repre- fented by the Fradlions — , — , jj, &c. This Hypothefis being admitted the foUov^^ing Corollaries may be deduced from it. Corollary I. The Probability of Life's failing in any Interval of Time AF is meafured by the Fradtion — . Corollary H. When the Interval ^F is once pafl, the Probability of Life's con- tinuing from F to G is j^ , for at F, the Complement of Life is FC S F, and the Probability of its failing is — . Corollary III. The Probability of Life's continuing from A to F, and then failing r -n . ^ • ^^ PG FG fromFtoG, is^X^.^^. T t 2 COROL- 324 Ihe Doctrine of Chances applied Corollary IV. The Probability of Life's failing in any two or more equal Inter- vals of Time affigned between A and S are exactly the fame, the Eftimation being made at A confidered as the prefent Time. Thefe things premifed, it will not be difficult to folve the following Problem. Two Lives being given, to find the Probabilty of one of them fixed lit on, furvivivi the other. A Bb S_ '\ I I I F Cc S '\ I I I For, let the Complements of the two Lives be refpedively AS=fT and FSi=p, upon which take the two Intervals AB, FC=:z, as alfothe two Moments Bb, Ccr=z. The Probability of the firft Life's continuing from A to B, or be- yond it, is -^^^ ; the Probability of the fecond's continuing from Fto C, and then failing in the Interval Cc, is by the third Corollary | : therefore the Probability of the firft Life's continuing during the time AB or beyond it, and of the fecond's failing juft at the end of that Time, is meafured by -^^ ^ J =^ ''~7/~ » whofe Fluent '"'~^ . 'i - will exprefs the Probability of the firft Life's continuing during any Interval of Time or beyond it, and of the fecond's failing any time before or precifely at the end of that Interval. Let now p be written inftead of z, and then the Probability of the firft Life's furviving the fecond, will be "^~^^^ r= i — -^ . From the foregoing Conclufion we may immediately infer that the Probability of the fecond Life's furviving the firft is ~ . By the fame method of arguing, we may proceed to the finding the Probabilityof any one of any Number of given Lives furviving all the reft, and thereby verifying what we have faid in Prob. XVIII. and XIX,. CHAPTER to the Valuation o/Annuities^ 325 CHAPTER IX. Serving to render the Solutions in this Treatife more ge- neral, and more correEi. I. Altho', in treating this fubjedl of Annuities, I have made ufe only of Dr. Halley's Table, founded upon the Brejlaw Bills of mortality j from which I deduced the Hypothefis of an equable Decrement of Life : Yet are my Rules eafily applicable to any other Table of Ob- fervations; by P7ob. II. of my Letter to Mr. yo?ies in Phil. Tranf-. N°. 473, which the Reader may fee below, in the Appendix. Or inftead of the Theorem there given, he may ufe that by which Trob. XXVI. was refolved, which is rather more independent of 'Tables : And its application to our prefent purpofe may be explain- ed as follows. As in all Tables of Obfervations deduced from Bills of mor- tality, or if we fhould combine feveral of them into one, it will be found that, for certain Intervals at leaft, flie Decretfients of Life con- tinue nearly the fame ; if we conceive the whole Extent of Life to be reprefented by a right Line AZ., in which there are taken diftances P^, ^R, RS, &c. proportional to thofe Intervals, and at the points P, ^, R^ S, &c. there be eredled perpendiculars propor- tional to the Numbers of the Living at the beginning of the re- fpedlive Intervals, and their Extremities are conne(fted by right Lines ; then there will be formed a Polygon Figure on the Bafe AZ, whofe Ordinates will every where reprefent the Numbers of that Table from which the Figure was conftrudted ; and the Inclinations of the Sides of the Polygon to its Bafe will exprefs the Convergencies of Life to its End, or the Degrees of Mortality belonging to the refpedive Intervals. Say therefore, as the difference of the Ordinates at P and ^^, is to the Ordinate at P: fo is the Interval P^, to a fourth PZ'; and PZ' fhall be the Complement of Life at the age P; and the Point Z' in the Bafe (hall be that from which the Complements are to be reckon- ed throughout the Interval P^ Let PZ', thus found, be fubllituted for n in the Canon o( Prob. XXVI, and the Interval P^for ;//, fo fhall the Value of that Interval be known : and in like manner the fubfequent Values of ^, RS, &c. giving to each Interval its proper Complement S>Z" , RZ"\ &c. And 2 26 Tloc DocTBi^iE oj Chances applied And laftly, thefe Values being feverally difcounted, Firjly in the Ratio of their refpetSive Ordinates at P, ^ i?, .&c. to fome preceding Ordinate as at N, at the Age I2, for inftance ; and Secondly, by ll^ie pr^ifent Value of i / payable after the Ypars denoted by NP^ M^, NR, &c. their Sum will be the Falue of the Life at N, ac- cording to the given Table of Obfcrvations. After which, the younger Lives muft be computed from Year to Year : as thofe after 70, or when an Interval contains but one Year, ought likewife to be computed. Jf it is propofed, for Example, to find how nearly my Hypotbefis agrees with Dr. Hallefs Table for the Interval of 8 Years between 33 and 41, it's Value, at 5 per Cent, computed by Prob XXVI. vi'ill, to an Annuitant 33 Years old, be 5.9456, according to the Hy- pofhc/is. But the Numbers of the Living at thofe Ages being, in the Table, 507 and 436, if we compute immediately from it, we muft take «r= ^—x 8 =57.14; and the fame Rule will give the Value 5.9831. Difcount now the Values found as belonging to a Life of 1 2 Years ,j that is multiply the firfl by -^ , and the other by -^~ ; and the Pro- duds 4. 2583 and 46957 difcounted tha fccond time, that is, multi- plied by .35H9, the prefent Value of i /. payable after 21 (=33 — 12) Years gives the Values 1.5283 and 1.6853; the difference being o. 1 5 J, near -T- of a Year's purchafe. In general, the Hypothcfis will he found to give the Value of a ftugle Life, or of an alligned Interval, iomewhat below what the Ta- ble makes it : but then, as both the young and the middle aged are obferved to die off" fafter in England than at Breflaw, my Rules may very well be preferable, for the Purchafes and Contracts tliat are made upon fingle Lives in this Country. In the fame mannei- may any other Tabki be compared with the Hypotbejisy and with one another. And if we give the preference to any particular Table, and would at the fame time retain the Hypo- tbefis of equal Decrement we may, by the diff'eretJtial Method, eafily find that mean Ternmiation of Life, Z, which fhall beft correfpond to the Table. n. To preferve fomewhat of Elegance and Ur>iformity in ray Solutions, as well as to avoid an inconvenient multiplicity of Camm and Symbols^ I did transfer the Decrement of Life from an Arithmeticd to a Geo- metrical to the Valuation o/" Annuities. 327 metrical Series : which however, in many Queftions concerning Combined Lives, creates an error too confiderable to be neglefted. This hath not efcaped tlie Gbfervation of my Friends, no more than it had my own : but the fame Perfons might have obferved likewife, that fuch Errors may, when it is thought neceflary, be correrted by my own Rule?; particularly upon this obvious princi- ple, That, if money 1 s fi'.ppofed to bear no Interejl^ theYzXnts of Lives 'will coincide iscith ivhat I call their Expedations. But as the Computation of fuch Corrcdions might feem tediousy and becaufe pradical Rules ought to be of ready Ufe, as well as fuffi- ciently exaft; I chufe rather to give another Rule hv joint Lives y which will anfwer both thefe Purpofes ; at the fame time that it is general, and eafily retained in the Memory. General Rule for the Valuation of joint Lives. 77}e given Ages being each increafed by unity ^.fmd^ by. Problem XXI. or XXII. the Ntitnber of Tears due to their joint Continuance ; and the Co7?ipUment of twice this Number to 86, taken as a piigle Life, will, iff the proper Table, give nearly the Value required. Example i. The Value of two joint Lives of 40 and 50, at 5 per Cent. waSj.r in Prob. II. found to be 7.62. But if they are made 41 and 51, their joint ExpeSiation, by Prob. XXI. will be 13 Years, thefe doubled and taken from 86 leave 60, againft which in Jable VJII, ftands 8.39 Years purchafe, nearly the Value fought. Exam pl e 2. The 3 joint Lives whofe fingle Values, at 4 per Cent, are 13, 14, 15 Years purchafe, are in Prob. II. worth 7.4 1. But by Table VJ, the Ages to which thefe Values belong, increafed by Unity, arc 42, 36, 28j whofe Complements to 86 fubftituted for />, 7/, ^, in: 7^ T^ + 77^7 ' *^^ Canon for the ExpeSlation of 3 joint Lives,, gives 12.43. And 86 — 2 X 12.43 is nearly 6i; at which Age a. fingle Life, in Table VI, is worth 8.75 Years purchafe. It is needlefs to add any thing concerning longeft Lives, Survivor- Mp^i Reverfom und Infurances , the Computation of tlaeir Values being; 328 7he Doctrine (?/" Chances applied &c. being only the combining thofe qI fmgk &nd joint Lives, by Addition and Subtradion : which being performed according to the Rules of this Treatife, the Anfwer may be depended upon as fufficiently exadl, in all ufeful Queftions that can occur. For we do not here aim at an Accuracy beyond what the determination of our main Data, the Probabilities of human Life, and the conformity of our Hypothefis to nature, can bear ; nor do we give our Conclufions for perfedlly exaft, as is required in fuch as are purely arithmetical, but only as ve- ry near Approximations; upon which bufinefs may be tranfafted, witiiout confiderable Lofs to any party concerned. in. The fame Rule ferves for the Cafe of an Annuify fecured, upon Joint Lives, by a Grant of Lands j or when the fradional part of the laft Year is to be accounted for. Only, in this Cafe, 1°. The Addi- tion of Unity to each Life is to be omitted. 2°, The fingle Life is not ■ — Ip now to be taken out of our Tables, or computed from -^— the Canon of Prob. I, but from -^^ -^P, a being Neper's Lo- garithm ofr : as in Phil. Tranf.'N°. 473, and in Chap. 1. foregoing. According to which, if the Ages and Intereft are as in Example i ; the ExpcSfation of joint Life will be 13.3 Years; and thence n=: 26.6 i P= 14.5358; ^ = .04879: And the Value of the Annuity 20 — 11.2 = 8.8 ; exceeding what it would have been upon yearly Payments by about -^ of a Year's purchafe. And if the Payments are half yearly or quarterly, the fkillful Com- putift cannot be at a lofs after what has been faid of thofe Cafes in Chap. VII *. * See, on the Subject of Annuities, Mathem. Repofuory., Vol. II. and III. by the ingenious Mr. James Dodfon, F. R. S. FINIS. appendix: N°. I. Dedication of the Firjl Edition of this Work (171 8.) TO SifTsfXAC Newton^ Kt. Prefident of the Royal Society. SIR, THE greateil Help I have received in writing upon this Sub- jedt having been from your incomparable Works, cfpecially your Method of Series ; I think it my Duty publickly to acknowledge, that the Improvements I have made, in the matter here treated of, are principally derived from yourlelf The great benefit which has accrued to me in this refpe(fl, requires my fhare in the general Tribute of Thanks due to you from the learned World: But one Advantage which is more particularly my own, is the Ho- nour I have frequently had oi being admitted to your private Con- verfation ; wherein the Doubts I have had upon any Subjeft relating to Mathematics, have been refolved by you with the greateft Hu- manity and Condefcenfion. Thofe marks of your Favour are the more valuable to me, becaufe I had no other pretence to them but the earneft defire of underftanding your fublime and univerlallv ufeful Speculations. I fhould think my felf very happy, if ha-, ing given my Readers a Method of calculating the Effe<fls of Chance, as thev are the refult of Play, and thereby fixing certain Rules, for eftimating how far fome fort of Events may rather be owing to Defign than Chance, I could by this fmall Efi"ay excite in others a defire of profe- cuting thefe Studies, and of learning from your Philofophv how to colledl, by a juft Calculation, the Evidences of exquifite'Wifdoni and Defign, which appear in the Phenomena of Nature throu<»hout the Univerfe. I am, with the utraoft Refped:, Sir, --■' -^ - Tour moft humble, and obedient Serva; U u A. de MoivRE. * ■ ' . Nofeu/on Goroll. i. ProU. Klli and upon Prob. IX. In that Corollary, it was found th at the P robabilities of winn ing all each others Stakes bemg as a? x^^— ^^ and bf y.ai — bf; If we divide by^ a — hy and, fuppofe the Chances for on? Gaijne to be equal, or a = b; then the Probabilities will be as the Number of pieces, or, in the Ratio of /> to q. Bnt when we ha,ve to. divide fuch Expreffions continually, that is by fonW Power ofiz'— >i as^^^r^S ^— ^^^^^ ^^- "^^ will be more convenient to ufe a General Rule for determining the Value of a Ra- tio whbfe Terms vanifli by the contrariety of Signs. The Rqlc IS thlS; For the difference of the ^antities that deflroy each other ip any Cafe, prop'jfed, write an indeterpiinate ^antity x ; in the Refult rejeSt all thofe' Terms tUt'v.a7iijf}ik-hen.jibecQmes.lefi, than any finite ^antity^: jofjall the remaining homogeneous Terms, divided by their greatefi com-, mon Meafure, exprej's the Ratio fought. As in our example, if we mak e a--b:=:x, or ar=b-\-x, and for a^, ai , write their equals b -\- xV, 3+^^ expanded by the Binomial^ Theorem; the Ratio of R to S, in Proi. VII, will be reduced tc^ that of pbi'+f-' X X +/. -^+/'? X bf+i-' X x^ + See. to qb^'+J-'. xx-j-g. ^~' X bf-+^~-Xx^-{-&ic: Of which retaining only the; two Terms thu involve x, and dividing them by bf+^-'xx, we get s. ? The Solution of Prob. IX. gives for the Gain of ^. tji§, Pro4ua> fubftitute as before, the Terms involving x vanifh in the Numerator of the firft of thefe Fadors ; reducing it to* *-f-7-X/' + yx ^/+f-» X X* + &c : and the Denom inator is * /> -j- ? X bf+^~' Xx ■\- &c. The other Fador is _^ii±i^±f^, or when x vanifhes with G-L refpcit to b, ^ i and the Produa: of the two is ^y X as^ ^i p p k N b i k: 5^{ as in Cafe 2. Cafe i follows immediately from this; and the ^d has as little difficulty. Another Example of our Rule may bej 'To Jind, from the Capon of Froh. I. of the Trea'tife on Annuities, the Expedation of a Life ivhoje Coifiplemefit is ii ; tiiat is, the prefeht Value of a Reyit or Annui- ty upon that Life, money bearifig tio Intereji. Now that Canon being : — , or — =^=- if for P we write" its equal -'~'" .- . and i A-k r — I nx'r— I •»• r — \ ' ' for r, the Value fought will be - — ^T^-/ — r= * * -j, 73T>; x**+ &c. n— I This Value wants half a year of ^ > its quantity according to the Rde given above, pag. aSBr^becaufe there rhe Probabilities of Life were luppofed to decreafe a the Ofdinates of a Triangle; whereas, in the Hypothefis of yearly payments Jri Prob, I, they decreafe feir faltum, like a Series of parallelograms infcribed in a Triangle. The Reader will likewife obferve that our general Rule for comput- ing the Value of a Fraction wliofe form becomes — , is iii effecfl the fame as that given by the Marquis de I'Hofpital in his Analyfe des infnimens petits. And that, from the Number of Terms that va- nifh in the Operation, and from the Sign of the Term which deter- mines the Ratio, the Species of algebraical Curve Lines, and the Pofition of their Branches, are difcovered. See Mac Laurin'i Fluxions, Book L Chap. 9. and Book IL Chap. 5. I^ u 2 N° III. 332 APPENDIX, N°. III. "Note to Prob, XLV.from Mr. Nicolas Bernoulli, Phil. Tranf. 341. To find the Probability that a Poule fliall be ended in a given Number of Games : a Series of Fradions beginning with 2 — I wbofe Denominators increafe in a double proportion, and the Nu- merator of each Fraction is the Sum of as many next preceding Nu- merators as there are Units in 71 — i, will give the fucceflive Pro- babilities that the Poule fliall be ended precijely in w, 7i-\-i, n-\-2, n-^-T,-, Sec. Games; and confequently if as many Terms of this Se- ries are added together, as there are units in p-\-i, their Sum will exprefs the Probability that the Poule (hall be ended at leaft in n -\-p Games. For Example, if there are 4 Players, and thence «=3, we Hiall have this Series ^ , j. -;V' 77 ' ■^' 717 ' 7^' TTI' ^^• Out of which if we form this other -L ^ 1. , ^ , -12. , ^, -^,. ,'°'- &c. whofe Terms are the Sums of the Terms of former Series, thefe laft will {hew the Probability of the Poule ending in 3, 4, 5, 6, &c. Games, at leaft. N°. IV. APPENDIX. ZIZ N°. IV A corre6i Table of the Sums of Logarithms me}it to his Mifcellanea 6.55976.30328.7678. I 46c 18. 38612. 46168. 7770. 470 32.42366.00749.2572. 48c 47.91164.50681.59^1. 49c 64.48307.48724.7^09. 5CC 81.92017.48493.9024. 51c 100.07840.50356.8004. 52c 118.85472.77224.9966. 53r 138. 17193. 57900. 1086. 54c 157.97000.36547.1585. 55c 17S. 2009 1. 76448. 7008. 56c 198. 825. 9.38472. 1977. 57c 219. 81069. 31561. 4815. 58c 241. 12910. 99886. 9689. 59c 262. 75689. 34109. 2616. 6ot 284.67345.62406. 8298. 61C 306.86078.19948.2847. 620 329. 3C297. 14247.9393. 63c 351.98588.98339.3535. 64c 374. 8968S.864CO. 4044. 65c 398.02458 26149 3624. 66c 421.35866.95421.3259 67c 444.88978-26514.6048. 6S0 468.6u936.87c56.479 1. 69c 492.50958.639-4.6190. 70c 516.58322.09826.1269. 71C 540.82361.20667.5295. 72c 565.22459.20470.1654. 730 589.7804.^.33690.9860. 74G 6i4.4858o.304c;7.7387. 750 639-33572-32255.0106. 760 66+. 32553. 6S/41. 5328. 77t 689. 45087. 77060. 382S. 78c 714.70764.378465691. 790 740.0919742162.3279. 800 765.6:022 85067.1998. 810 791.22896.82108 465S. 820 816.97493.05636.3600. .830 842 8^506.30337.0506. S40 868. 80641.41777. 2588. 85c 894.8862138085.1630. 860 921.07182.03166.5465. 87 947.36071.70083.752(5. 8S0 973-7j050.41416.4i85. 890 1000.23889.09583.9930. 900I. )C 20 30 40 50 60 70 80 9' 100 1 10 120 130 140 150 160 170 180 190 200 210 220 230 240 250 260 280 290J 300 310: 320] 330, 340 350 360 370, 380 390 400 410 420 430 440, 450; , from the Author's Supple- Analytica. 1026 82368.84245.7267. 1 053. 50280. 2 600^.6230. 1080.274:2 85779.2496. 1107.13604.49151.6763. 1 134.08640.85351. 35o8i 1161.12355.00246.5923. 1 188 24576.93048.6770. i2i545i43-ib339-625i. 1242.73S96.391 14.8380- 1270. 106 5 12561.5931. J297-55363-38324.8209. 1325.07790.39038.2121. 1 352.67830.3092 2.049 1. 13S035331 98269.6983.. 1408.1 02 28.69662.7 808. •435-92337 9.'^77i-' »H. 1463 8 1 561.28607.3923. 1491. 77784.02119. 6951. 1519 8oS95.r40i5 3428. 1547 90787.08720.1888. 1576.07355.61385.9540. 1604.30499 62866.2770. 1632.60 1 2 1.05589.2142. j66o 96124. 70260.3147. 1 689.38418.13336.1091. 1717.86911.552130134. 1746.41 5 17. 6908 1. 2925. 1775.02151. 70397.9157- 1S03. 6873 1.06935.9463. 1832. 41175.49371. 5144- 1861.19406.82-5^2.5655. 1890:03 ^48.96156 3791V' 191892927.78485.4396. 1 947. 8 807 1. 07073. 5663. 197&.S8708. 42376.3542. 2005.9477 1.20 74 1.9 1 5 2. 2035.06192 47899.6883. 2064.22906 92766. 7i82> 2093.44S50 81552.2793. 2122.71961.92143.1027. 2 1 52. 041 79 48752. 701 3- 2181.41444, 1 6s 19 4477 .-■ 2210.8:1697.98139.1145/ ' 2240.30884.26218.5633.- 2269. 82947. 618 38. 1 577,. 3^;+ APPENDIX. If we would examine thele Numbers, or continue the Table farther on, we have that excellent Rule communicated to the Author by Mr. James Stirlhig ; publiflied in his Supplement to the Mifcellanea Anaktica, and by Mr. Stirling himlelf in his Methodus Diferentialis, Prop! XXVIII. " Let z — {-be the lail Term of any Series of the natural Num- *' bers I, 2, 3, 4, 5 ^ — 1; '^ =-43429448 '90325 thereci- *' procal of Neper's Logarithm of lo : Then three or four Terms of " tJiisSerieszLog.z — ^2:— ^-^-|-~^ ^TTT^p- + 77173^7 " — &c. added to 0.399089934179, &c. which is half the Loga- *' rithm of a Circumference whole Radius is Unity, will be the Sum «' of the Logarithms of the given Series ; or the Logarithm of the " Produdt 1x2x3x4x5 xz 1" The Coefficients of all the Terms after the firft two being formed as follows. Put —-^ = A = ^+iojB + 5C = ^-l-2i5-l-35C-l-7£) ,,^^=^+36£-f i26C+84D+9£;. &c. In which the Numbers i, i, I, Sec. 3, 10, 21, 36, &c. 5, 35, 126, &c. that multiply A, B, C, 6cc. are the alternate Uncia of the odd Powers of a Binomial. Then the Coefficients of the feveral Terms wmbe^x^=— ^,T^X^=8i?.ll'x^=ItTl6^.&^- See the general Theorem and Demonftration in Mr. Stirling's PrOpo- lition quoted above. N". V. Some Vfeful Cautions. One of the mofl frequent occafions of Error in managing Pro- blems of Chance, being to allow more or fewer Chances than really there are ; but more especially in the firft Cafe, for the fault lies com- monly that way, I have in the Introdudion taken great care to fettle the Rules of proceeding cautioufly in this matter ; however it will not be amifs to point out more particularly the danger of being miftafccn. Suppofc 7- '2 I 9.16 I APPENDIX. n,z.^ Suppofe therefore I have this Queftion propofed ; There are two Parcels of three Cards, the firft containing King, Queen, and Knave of Hearts, the fecond the King, Queen, and Knave of Diamonds, and that I were promifed the Sum S, in cafe that in taking a Card out of each Parcel, I fhould tai^c out either the King of Hearts, or tlie King of Diamonds, and that it were required I fliould determine the value of my Expectation. If I reafon in tWs manner j the Probability of taking out the King of Hearts is — , therefore — / is my due upon that account j the Probability of taking out the King of Diamonds is alfo — , and therefore that part of my Expedlation is — / as the other was, and confequently my whole Expedlation is — /; this would not be a le- gitimate way of reafoning : for I was not promifed that in cafe I fhould take out both Kings, I fliould have the Sum if, but barely the Sumy.' Therefore we muft argue thus j the Probability of taking out the- King of Hearts- is — , the probability of mifllng the King of Diamonds is — , and therefore the probability of taking out the King of Hearts, and miffing the King of Diamonds is -^ x — = — , for which reafon that part of my Expectation which arifes from the probability of taking out the King, of Hearts, and milfmg the King of Diamonds is — /; for the fame reafon that part of my Expedlation which arifes from the probability of taking the King of Diamonds and miffing the King of Hearts is — y, but I ought not to be deprived of the Chance of taking out the two Kings of which the probability is — and therefore the value of that Chance is —f; for which reafon, the v^lue of my whole Expedlation is -/-j- -/+ -/= -/ which is lefs by ^y than -f. But fuppofe I were propofed to have if given me in cafe I took out. both Kings, then this lafl Expcdation would-be -/,' which wpuld make the whole value of my Expedation to be -f -\- One 33^ A P P E N D I X. One may perceive by this fingle inftance, that when two Events are fuch, that on the happening of either of them I am to have a Sum f, the probability of that Chance ought to be eftimated by the Sum of the ProbabiUties of the happening of each, wanting. the pro- bability of their both happening. But not to argue from particulars to generals. Let x be the pro- bability of the happening of the firfT:, and y the probability of the happening of the fecond, then a: x i — y or x — xy will reprefent the probability of the happening of the firfl: and failing of the fe- cond, and J X 1 — X or^' — xy will reprefent the probability of the happening of the fecond and failing of the firft, but xy reprefents the happening of both ; and therefore x — xy-\-y — xy-\-xy ov X _|- y — xy will reprefent the probability of the happening of either. This conclufion may be confirmed thus ; i — x being the pro- bability of the firft's failing, an d i — y the p robability of the fe- cond's failing, then the Produd i — a; x i — ;- or i — x — y-\-xy will reprefent the probability of their both failing ; and this being fubtra6ted from Unity, the remainder, viz. x-{-y — ^9' will repre- fent the probability of their not both failing, that is of the happen- ing of either. And if there be three Events concerned, of which the Probabili- ties of happening are refpedively x, y, z, then multiplying i — x by , y and tliat again by i — z, and fubtrading the Produdl from Unity, the remainder v-'ill exprefs the probability of the happening of one at lead of them, which confequently will be x -\- y -^ z ~ xy xz yz-{- xyz ; and this may be purfued as far as one pleafes, A difficulty almofl: of the fame nature as that which I have ex- plained is contained in the two following Quellions : the firfl: is this ; A Man throwing a Die fix times is promifed the Sum /every time he throws the Ace, to find the value of his Expedation. The fecond is this; a Man is promifed the 6um/if at any time in fix trials he throws the Ace, to find the value of his Expeda- tion. In the firfl: Queftion every throw independently from any other is entitled to an Expedation of the Sum/ which makes the value of the Expedation to be |/+ f/+ 7/-]- 7/+ 7/+ 7/= A but in the fecond, none but the firfl throw is independent, for the fecond has no right but in cafe the firfl has failed, nor has the APPENDIX, -ixi the third any right but in cafe the two firft have failed, and fo on ; and therefore the value of the Expedlation being the Sum ex- pected, multiplied by the Sum of the Probabilities of the Ace's be- ing thrown at any time, exclufive of the Probabilities of its having been thrown before, will be -r f '\ — 7~ f -\ ~fA -^—^■\- -^fJr -^/= ^H- that is nearly i-/ 7776'' ' 4't);6-' 46.'';f) •' 3'' We may alfo proceed thus -, the probability of the Ace's being miffed fix times together is— X-fX— Xt-Xt-Xt = —rr~^ > 060000 4*()> ' and therefore the probability of its not being miffed fix times, that is of its happening fomc time or other in 6 throws is 1 .fo a = 4^^ , and confequently the value of the Expedtation is ]'^^.'.^ - as it was found before. Another Inflance may be, the computing the Odds of the Bet, That one of the 4 Players at Whifl JJ^all have alcove 4 Trumps. The Solution one might think was by adding all the Chances (in the Ta- bles pag. 177) which the 4 Gameflcrs have for 5 or more Trumps ; and this would be true, were every Gamefler to lay for himfclf in particular. But as it may happen that two of the Gameflcrs have above 4 Trumps, and yet, as the Bet is commonly laid, only one Stake is paid, half the Number of thefe lafl Chances (computed by Prob. XX.) is to be fubtradted : which reduces the Wager nearly to an equality. N°. VI. AJhort method of calculating the value of Annuities on Lives, from Tables ofObfervations, In a Letter to W. Jones Efi; Phil. Tranf. N\ 473. Although it has been an eftablifhed cuflom, in the payment of Annuities on Lives, that the lalt rent is lofl to the heirs of the late poffeflbr of an annuity, if the perfon happens to die before the ex- piration of the term agreed on for payment, whether yearly, half- yearly, or quarterly : neverthelefs, in this Paf>er I have fuppofed, that fuch a part of the rent Ihould be paid to the heirs of the late pofTeffor, as may be exadly proportioned to the titne elapfed between that of the lafl payment, and the very moment of the Life's expir- ing ; and this by a proper, accurate, and geometrical calculation. I have been induced to take this method, for the following rea- fons ; firft, by this fuppofition, the value of Lives would receive but X X an 338 APPENDIX. an inconfiderable increafe ; fecondly, by this means, the feveral inter- vals of life, which, in the Tables of Obfervations, are found to have unifomi decrements, may be the better connected together. It is with this view that I have framed the two following Problems, with their Solutions. PROBLEM I. To find the value of an Annuity, fo circumftantiated, that it JJoall be on a Life of a given age ; and that upon the jailing oj that lije, fuch a fart of the rent pall be paid to the heirs of the late pojfejfor of an An- nuity y ai may be exaBly proportioned to the time intercepted between that of the lafl payment ^ and the very moment of the li fe' s failing. Solution. Let n reprefent the complement of life, that is, the interval of time between the given age, and the extremity of old-age, fuppofed at 86. r the amount of i /. for one year. a. the Logarithm of r. P the prefent value of an Annuity of i /. for the given time, ^the value of the life fought. Then — - = ^ r — I an ^^ Demonstration. For, let 2; reprefent any indeterminate portion of w. Now the Pro- bability of the life's attaining the end of the interval z, and then fail- ing, is to be expreffcd by - , (as {hewn in my book of Annuities upon Lives) upon the fuppofition of a perpetual and uniform decre- ment of life. But it is well known, that If an Annuity certain of i /. be paid J during the time 2;, its prefent value will be P = - y_^ - or — — — ; — I xr And, by the laws of the Dodrine of Chances, the Expeftation of fuch a life, upon the precife interval z, will be exprcffed by Z—- — — ^=i ; which may be taken for the ordinate of a curve, whofe area nr*x.r — I is as the value of the life required. In. APPENDIX. 339 In order to find the area of this curve, let/>=«xr — i ; and then the ordinate will become y — ~p ■> ^ much more commodious expreflion. Now it is plain, that the fluent of the firft part is y ; but as the fluent of the fecond part is not fo readily difcovered, it will not be im- proper, in this place, to fhew by what artifice I found it ; for I do not know, whether the fame method has been made ufe of by others : all that I can fay, is, that I never had occafion for it, but in the parti- cular circumftance of this Problem. Let, therefore, r' = xj hence 2; Log. r = Log. x ; therefore z Log. r = (Fluxion of the Log. x =) j , or a z := — ; confequently and -4- = -^ : but the fluent of -^ is ( ''— z=) r'- axx a.xx ^ ax ' X »x ' r- axx and therefore the fluent of — -^ will be 4- fr^ ' foe,'- The fum of the two fluents will be T + T^; ^^^> whenz = o, the whole fluent fhould ber=o; let therefore the whole fluent be ^ p Now, when 2 = 0, then— = 0, and— becomes — (for r*=i,) confequently -^ -\- q=:o ; and qz=. -: therefore the area of a curve, whofe ordinate is ~ will be f — A — = \ - p fr "^p ay ' «,y* / /, I I I r X — . r a,p But P = — — '- , ; therefore i L = r — i x P, and the expreflion for the area becomes ■ 1 — — — : And putting n in- (lead of z, that area, or the value of the life, will be expreffed by — ! . ^E. D. Thofe who are well verfed in the nature of Logarithms, I mean thofe that can deduce them from the Dodrine of Fluxions and infi- nite Series, will eafily apprehend, that the quantity here called «, is that which fome call the hyperbolic Logarithm ; others, the natural Logarithm : it is what Mr. dUs calls the Logarithm whofe modulus is I : laftly, it is by fome called Nepers Logarithm. And, to fave the reader fome trouble in the pradice of this laft theorem, the moft: necefl'ary natural Logarithms, to be made ufe of in the prefent dif- quifition about Lives, are the following : X x 2 If 340 APPENDIX. If r = I. 04, then will a = o. 0392207. r=1.05, - - - a r=: o. 0487901, r = i.o5, - - - a ^= o. 0582589. It is to be obferved, that the Theorem iiere found makes the Values of Lives a little bigger, than what the Theorem found in the firft Problem of my book, of Annuities on Lives, does ; for, in the prefentcafe, there is one payment more to be made, than in the other ; however, the difference is very inconliderable. Bur, although it be indifferent which of them is ufed, on the fuppofition of an equal decrement of life to the extremity of old-age ; vet, if it ever happens, that we fhould have Tables of Obfervations, concerning the mortality of mankind, intirely to be depended upon, then it would be convenient to divide the whole interval of life into fuch fmaller intervals, as, during which, the decrements of life have been obferved to be uniform, notwithftanding the decrements in fome of thofe intervals fhould be quicker, or flower, than others j for then the Theorem here found would be preferable to the other ; as will be fhewn hereafter. That there are fuch intervals, Dr. Hallefs Tables of Obfervations fufficiently fhew j for inflance ; out of 302 perfons of 54 years of age, there remain, after 16 years ('that is, of the age of yo) but 142 ; the decrements from year to year having been conflantly 10 ; and the fame thing happens in other intervals ; and it is to be prefumed, that the like would happen in any other good Tables of Obfervations. But, in order to fliew, in fome meafure, the ufe of the preceding Theorem, it is neceffary to add another Problem; which, though its Solution is to be met with in the firfl edition of my book of Annui- ties on Lives, yet it is convenient to have it inferted here, on account of the connexion that the application of the preceding Problem has with it. In the mean time, it will be proper to know. What part of the yearly rent Jhould be paid to the heirs of the late pofeffor of an Annuity^ as may be exaBly proportioned to the time elapfed between that of the lafl payment^ and the very moment of the life's expiring. To determine this, put A for the yearly 'ent; '- for the part of the year intercepted between the time of the laft payment, and the inflant of the life's fail- ing ; r the amount of 1 1 at the year's end.: then will \Z\" ^ ^^ *^^ fum to be paid^ PRO- APPENDIX. 341 P R O B L E M II. To find the Value of an Annuity for a limited interval of life, during which the decrements of life may be cot fide red as equal. Solution. Let a and b represent the number of people living in the beginning and end of the given interval ot years. s reprefent that interval. P the Value of an Annuity certain for that interval, ^the Value of an Annuity for life fuppofed to be necclTarily cx- tinft in the time s; or (which is the fame thing) the Value of an Annuity for a life, of which the complement is s. Then ^-| — x P — ^^will exprefs the Value required. Demonstration. For, let the whole interval between a and b be filled up with arith- metical mean proportionals ; therefore the number of people liv- ing in the beginning and end of each year oft' e given interval s will be reprefented by the following Series ; viz. a . -1— . . — . C^c. to b. S S i s Confequently, the Probabilities of the life's continuing during r, 2, 3, 4, 5, ^c. years will be exprefled by the Series,' . ————— , . . isc. to — , .a J'i sa sa a Wherefore, the Value of an Annuity of \ I. granted for the time J^ will be exprefled by the Series sa — n-^h . sa — zn-'rib ^ sa — % -\xh |^ sa — 40+ li -,. . /• this Series is divifible into two other Series's, n^iz. J sr ' sr'' ' sri ' s'* 'si' 2d. i- X— -1--^+ -V-r-T, &c.to-^. a sr ' s>- ' sii ' sr* ' J. ' Now, fince the firft of thefe Series's begins with a Term whofe Numerator is s — l, and the fubfequent Numerators each decrcafe by unity ; it follows, that the lafl: Term will be = o ; and confequent- ly, that Series exprefl"es the Value of a life necefllxrilv to be ex- tindl in the time s. The fum of which Series may be efleemed as a given quantity J and is wh^ I have exprefled by the fymbol ^\r. Problem i. The 342 APPENDIX. The fecond Series is the difference between the two following Series's,_ Where, neglecting the common multiplier - , the firll Series is the \'alue of an Annuity certain to continue s years; which every mathe- matician knows how to calculate, or is had from Tables already com- pofed for that purpofe : this Value is what I have called P j and the fecond Series is ^ Therefore ^+ - X P— ^will be the Value of an Annuity on a life for the limited time. ^ E. D. It is obvious, that the Series denoted by ^ muft of neceflity have one Term Icfs than is the number of equal intervals contained in s ; and therefore, if the whole extent of life, beginning from an age '^iven, be divided into feveral intervals, each having its own particu- lar uniform decrements, there will be, in each of thefe intervals, the defed of one payment ; which to remedy, the Series ^muft be cal- culated by Problem i . Example. Tojind the Value of an Annuity for an age of 54, to continue 16 years^ and no longer. It is found, in Dr. Halleys Tables of Obfervations, that a is 302, and b ijz: now « z= ^ = 16 ; and, by the Tables of the Values of Annuities certain, P^io.^ijy ; alfo (by Problem 1.) ^= ('^ — — z=) 6.1 168. Hence it follows (by this Problem), that the Value of an Annuity for an age of 54, to continue during the limited time of i6 years, fuppofing intereft at 5 per cent, per annum, will be worth {.^-V T X ^— ^— ) ^-3365 years purchafe. From Dr. Hallefs Tables of Obfervations, we find, that from the age of 49 to 54 inclufive, the number of perfons, exifting at thofe feveral ages, are, 357, 346, 335' 324, 3^3' 3^2, which compre- hends a fpace of five years ; and, following the precepts before laid down, we fliall find, that an Annuity for a life of 49, to continue for the limited time of 5 years, intereft being at 5 per cent, per an?mrn, is worth 4.0374 years purchafe. And, APPENDIX. 343 And, in the fame manner, we (hall find, that the Value of an An- nuity on a life, for the limited time comprehended between the ages of 42 and 49> is worth 5.3492 years purchafe. Now, if it were required to determine the Value of an Annuity on life, to continue from the age of 42 to 70, we muft proceed thus: It has been proved, thgit an Annuity on life, reaching from the age of 54 to 70, is worth 8.3365 years purchafe; but this Value, being eftimated from the age of 49, ought to be diminiflied on two accounts: Firft, becaufe of the Probability of the life's reaching from 49 to 54, which Probability is to be deduced from the Table of Obfervations, and is proportional to the number of people living at the end and beginning of that interval, which, in this cafe, will be found 302 and 357: The fecond diminution proceeds from a dif- count that ought to be made, becaufe the Annuity, which reaches from 54 to 70, is eftimated 5 years fooner, viz. from the age of 49, and therefore that diminution ought to be expreffed by -^ ; fb that the total diminution of the Annuity of 16 years will be expreffed by the fradion -^^^ , which will reduce it from 8.3365 years pur- chafe to 5-5259; this being added to the Value of the Annuity to continue from 49 to 54, viz. 4.0374, will give 9-5633, the Value of an Annuity to continue from the age of 49 to 70. For the fame reafon, the Value 9-5633, eftimated from the age of 42, ought to be reduced, both upon account of the Probability of living from 42 to 49, and of the difcount of money for 7 years, at 5 per cent, per annum, amounting together to 3-8554, which will bring it down to 5- 7079 ; to this adding the Value of an Annuity on a life to con- tinue from the age of 42 to 49, found before to be 5.3492, the fum will be 1 1. 0571 years purchafe, the Value of an Annuity to continue from the age of 42 to 70. In the fame manner, for the laft 16 years of life, renching from 70 to 86, when properly difcounted, and alfo diminifhed upon the account of the Probability of living from 42 to 70, the Value of thofe laft 16 years will be reduced to 0.8 ; this being added to 1 1.057 1 (^^^ Value of an Annuity to continue from the age of 42 to 70, found before), the fum will be 11.8571 years purchafe, the Value of an Annuity to continue from the age of 42 to 86 ; that is, the Value of an Annuity on a life of 42 5 which, in my Tables, is but 1 1.57, upon the fuppolition of an uniform decrement of life, from an age given to the extremity of old-age, fuppofed at 86. 344 APPENDIX. It is to be obferved, that the two diminutions, above-mention- ed, are conformable to what I have faid in the Corollary to the fecond Problem of the firfl edition, printed in the year 1724. Thofe who have fufficient leifure and fkill to calculate the Value of joint Lives, whether taken two and two, or three and three, in the fame manner as I have done the firft Problem of this tradl, will be greatly aflifted by means of the two following Theorems : If the ordinate of a curve be -4-; its area will be -rr— ?;r — If the ordinate of a curve be -^j its area will be ~ — ^ — N^ vii. APPENDIX. N°. VII. The Probabilities of human Life, accordifig to different Authors. Table I, by Dr. Hallcy. 345 Age iving. Age 1 iving. 1 31 1 Living. , 523I A£,e 46 Livinc. 387 ^ge Living. Age 76 I iving. I 78: i lOO' i6 622 61 232 2 89s '7 61632 515! 17 377 62 222 '7 68 3 79 18 61033 5o;i + 8 367 63 212 7« 58 4 76c 19 60434 499 +9 357 /-)4 202 79 ■' 49 9 73^ .0 59835 *49o s:o *346 ^5 'I 80 4' 6 7 10 21 59236 481 51 335 06 18. 11 34 7 69. 22 586|37 472 - 3^4 57 17? '2 28 8 680 23 580 3" 46? i? 3 '3 68 162 ^3 23' 9 67c 574 39 45. :4 3°^ 69 152 H 19 lO 66 ^- *56- +0 44 > ss *292 70 14^ * * 1 1 6;; 6 560 41 43" _ u 282 7^ *,3, ' 2 6+c 27 553 42 427 57 27; 72 I2C '3 *6 f. b 546 43 *4>7 5« 267 73 lOy ^4 ^^3 ^9 539 4-.1 407 '9 252 74 9'' icj 6 *:.3' K 397 6c 24: 75 * ' Table II. by M. Kcrjfeboom. \ge; Living. 11 1 0^1400 Age 76 Living. 849 Age Living. \ge 1 iving. Age 61 Living. 365 7^ Living. 160 JA.e 91 Livine. \ 7 1 1.2^1 3' 699, 46 550 2 'O7.5! '7 842 32 687' 47 540 !62 356 71 145 92 5 3 1030; 18 855 33 ^75, 48 530 h 343 7l- 130 93 6 4 993 •9 826 34 665 49 518 ,64 329 79 1^5 94 2 9 964 20 817 35 655 50 S^7 '66 3^5 :c 100 95 I i 6 947 2 1 808 36 645 51 495 301 81 87 96 0.6 7 930 22 800 37 635: 52 482 1^7 287' •>'2 75 97 0-5 8 913, 23 792 38 625 ^^l 470 ,68 273 ^3 64 98 0..', 9 904 M 783 19 615: 54 458 I99 259 84 SS 99 0.2 10 895 25 772 40 605 S5 446 |7° 245I 85 45 100 0.0 1 1 1 8H6 26 76c 41 596 56 434 i7' 231! 86 36 ^ 12 8781 27 747 42 587 57 421 7^ 217 «7 28 13 28 735 y^ 578 58 408 I73 203 88 21 H 863; '9 723 44 569 59 395 174 189 89 ^5 K ^ 6 x^- 711 45 560 60 382 \7^ ^7'^. 90 ]0 Tabic II L 34^ A P P E N D I X. Table III. by M. de Tar deux. Ag<- 1 . iving — Ag i-iving Agf Living. Ag. Living. Age Living. 1 **** 2 J 806 4' 650 61 450 81 101 2 «*** 22 79^ 42 643 62 437 82 85 n ^ 1000 23 79c +3 636 63 423 «3 7' L 97c 24 782 44 629 64 409 84 59 S 94H 25 11'. 45 622 65 395 «s 48 L 93° 26 766 L 615 66 380 86 38 7 9M 27 IS"^ ^7 607 67 364 87 29 < 1 902 2\ 75^ 48 599 68 347 88 22 9 890 29 742 49 59° 69 329 89 16 IC 880 30 734 5° 581 70 310 90 II I i 872 31 726 5' 51' 7' 291 91 7 12 866 >2 71^ 52 560 72 271 92 4 •3 860 33 71C 53 549 73 251 93 2 4 854 34 702 54 53« 74 231 94 I 15 848 3S 69^ 55 526 15 21 1 95 If 842 36 686 5t 514 76 192 96 * '7 «35 37 678 S7 502 77 '73 97 * i5- 828 3« 671 5« 489 7^ 154 98 '^ 821 39 664 59 476 79 136 99 2.| 814 40 657 6c 463 So iiH lool 1 Table IV by Meflieurs Smart and Simpfon. Afc Living, AgE Living. Age Living. Age Living. •4ge Living, 1 12801 8705 ''. 480 33 358 49 212 65 99 2 700 18 474 34 349 50 204 66 93 3 635 19 468 35 340 51 196 67 87 4 600 20 462 36 331 52 188 68 81 5 580 21 455 yj 322 53 180 69 75 6 564 22 448 38 3^3 54 172 70 69 7 551 23 441 39 304 55 ,65 71 64 8 541 24 434 40 294 56 158 72 59 9 532 25 426 41 284 51 15' 11 54 10 524 26 418 42 274 58 144 74 49 1 1 517 27 410 43 264 59 137 75 45 12 5JO 28 402 44 255 60 13' 76 41 13 504 29 394 45 246 6] 12 17 38 '4 498 30 385 46 237 62 117 y'b 35 M 492 31 3t6 47 228 6^, II] 79 32 16 4'<6 32 367 48 22c 64 105 |8o 29 Remarks APPENDIX. 347 Remarks on the foregoing Tables. The firfl: Table is that of Dr. Halley, compofed from the Bills of Mortality of the City of Brejlaia ; the beft, perhaps, as well as the firft of its kind ; and which will always do honour to the judgment and fagacity of its excellent Author. Next follows a Table of the ingenious Mr. Kcrffeboom, founded chiefly upon Regifters of the Dutch Annuitants, carefully examined and compared, for more than a century backward. And Monfieur de Parcieux by a like ufe of the Lifts of the French Tontines, or long Annuities^ has furniflied us Table IH; whofe numbers were likewife verified upon the Necrologies or mortuary Regifters of feveral religious houfes of both Sexes. To thefe is added the Table of Meflieurs Smart and Simpfon, adapt- ed particularly to the City of Lo7idon ; whofe inhabitants, for reafons too well known, are fhorter lived than the reft of mankind. Each of thefe Tables may have its particular ufe : The Second or Third in valuing the better fort of Lives, upon which one would chufe to hold an Annuity j the Fourth may ferve for London, or for Lives fuch as thofe of its Inhabitants are fuppofed to be : while Dr Halley'^ numbers, falling between the two Extremes, feem to approach near- er to the general courfe of nature. And in Cafes of combined Lives, two or more of the Tables may perhaps be ufefully employed. Befides thefe, the celebrated Monfieur de Biiffon -^ has lately given us a new Table, from the adtual Obfervations of Monfieur du Pre de S. Maur of the Fre^ich Academy. This Gentleman, in order to ftrike a juft mean, takes three populous parifhes in the City of Paris, and fo many country Villages as furnifh him nearly an equal number of Lives: and his care and accuracy in that performance have been fuch as to merit the high approbation of the learned Editor. It was therefore propofed to add this Table to the reft ; after having purged its numbers of the inequalities that neceflarily happen in fortuitous things, as well as of thofe arifing from the carelefs manner in which Ages are given in to the parifh Clerks ; by which the years that are multiples of lo are generally overloaded. But this having been done with all due care, and the whole re- duced to Dr. Hallefi Denomination of icoo Infants of a year old ; there refulted only a mutual confirmation of the two Tables ; Mr. du Prfs Table making the Lives fomewhat better as far as 39 years, and thence a fmall matter worfe than they are by Dr. Halley's. We may therefore retain this laft as no bad ftandard for mankind in general; till a better Police, in this and other nations, ftiall furnifh f Hijioire NaturalJe, tome II. the 348 APPENDIX. the proper Data for corre(Si:ing it, and for exprefling the Decrements of Life more accurately, and in larger numbers. For which purpofe, the parifh Regifters ought to be kept in a better manner, according to one or other of the Forms that have been pro- pofed by Authors. Or, if we fuppofe the numbers annually born to have been nearly the fame for an age part, the thing may be done at once, by taking the numbers of the living, with their ages, through- out every Parifli in the Kingdom : as was in part ordered fome time ago by the Right Reverend the Bifhops: but their Order was not uni- verfally obeyed ; for what rcafon we pretend not to guefs. Cert, n it is, that a Cetifiis of this kind once eftablifhed, and repeated at proper in- tervals, would furnifh to our Governours, and to ourfelves, much im- portant inftruftion of which we are now in a great meafure deftitute : Efpecially if the whole was diftributed into the proper Clajfes of mar~ ried and u?i»/arriecly indnjlrioui and chargeable Poor, Artificers of eve- ry kind, ManufaBurert^ &c. and if this was done in each County, City, and Borough, feparately ; that particular ufeful conclufions might thence be readily deduced; as well as the general ftateof the Nation difcovered; and the Rate according to which human Life is wafting from year to year. See, on this fubjedl, the judicious Ob- fervations of Mr. Corbyn Morris, addrefled to Thomas Potter Efq; in the year 1751. F J N I S. d" UNIVERSITY OF CALIFO Los Angel This book is DUE on the h ^ 3§R''>J^M JUN281971 at -; Form L9-Series 4939 ^9^ MAY 3 1 1995 \D ^.t OA 273. M S'^ 3 1 58 00297 0316 UC SOUTHERN R? D 000 829 475 3 ■ejLuia>fii UNIVERSITY OF CALIFORNIA LIBRARY Los Angeles This book is DUE on the h ^ 5§f)''^fe°lSH JUN281971 ?5^ fit JAN^ Form L9-Serics4939 m WlAY vo .«!<». ,^-^1.1^^ OA 273. M s^ 3 11 58 00297 0316 D 000 829 475 3 ■■.k^aiitias L