The Doctrine of Chances, Or, A Method of Calculating the Probability of Events in Play

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STANFORD UN(VEBSITV- LIBRARIES

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O R,

A Method of CalGulacing r'le Probability
of Events in Play.

By A. Be Mohre. F. R. S.

£ N D N:
VmtedbyW.Pearfm, for the Amlior. MDCCXVllI,

■

• .• • •. • . •

.••■•-.••.••;;••• • •. • , •

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<- %-N.

•««•••*>»««•• <

On

T O

in vac Newtoiu Kl Prejideut

of the Hoyai Society,
s I K

HE greateft help I have re-
ceiv a in writing iipon this Sub-
jciSt having been from your In-
comparable Works, efpecially your
Method of Series j I think it my Diity
publickly to acknowledge, that the
Improvements I have made in the
matter here treated of, are principally
derived from your ie\£. The great
benefit which has accrued to me in
this refped:, requires my (hare in the
general Tribute of Thanks due to
you from the Learned World; But
one advantage, which is more parti-
cularly my own, is the Honour 1 have
frequently had of being admitted to
your private Converfation, wherein
the doubts I have had upon any Sub-
je6: relating to Mathematics ^ have been
refblved by you with the greateft
Humanity and. Condefcention. Thoie
: . . . Marks

609839

Hbf Dedication,

Marks of your Favour are the more
valuable to me, becaufc 1 had no other
pretence to them, but tlie eanicil- dc-
lire of und: "{landing your ial-MmQ
and univerJaily uleful Spx i!- i:»ons.
I fhould think my felf very ha|)py, if,
having given my Readers a Method of
calculating the Effeds of Cliance, as.
they are the refult of Play, and thereby
lix*d certain Rules, for efiimating how
far fbme fort of Events may rather be
owing to Deiign than Chance, I could
by this fmall Eflay excite in others a
defire of profecuting thefe Studies,
and of learning from your Philofbphy
how to colled, by a juft Calculation,
the Evidences or exquifite Wifdom
and Defign, which appear in the Fbe^'
nomefia of Nature throughout the
Univerfe. I am, with the utmoft
Refped,

Sir,

Your moft Humble,

and Obedient Servant,

A De jMoivre.

(i)

y^yc '^y-M< V.'Vif X^^^ 'i>it^< V^'^( ^'*:

PREFACE.

'^>l"'"^/5 A-oic lio/ir JfTW rc'.iJ'/, jf»« 7 f^xvt A Sftcimm
H /» t/jc Pinlofophical Tranfadioiis, tf wbdt I
[l fioiv mare hrgtlj treat ef in tbit Book. The

-^^ ottafion cf mj then uitJertdti/tg tbit SutyeS trtu
cbiefij owing to the Defire Mi Emoartgement <f the Ho^
nourahU Mr. Francis Robartes, ■>>£#, ufon octagon tf m
French Tra£fyca3edf L'Analyfe desjcux de Hazard, ip/j«A had
lately iitn PahiifkeJf tfot fleaftA to frofofe to me feme Pro-
Hems of much greater di^cuUy than any hi had found in that
Book; which having fotved to bis Saiiifa^iony he engaged site
to Methodife thofe Problems^ and to lay dotvn the Rules iphith
had ted me to tUir Solution. After I had proceeded thus fgr^
it was enpintd me ty the Royal Society, to communicate to ibem
jphat I had difcovtrei on this Sabjkf, and thereupon it wdt
ordered to be pab/rjhed in tbe TranfaiHons, not as a matter re-
lating only to Play, but at containing fome gentral Steculationt
not unworthy to he eonjidered by the Lovers of Truta.

I had not at that time read any thing concerning this Snb-
jeil, but Mr. Huygens'j Book, dc Ratiocinits in Ludo Alex,
and a little Engli/h Piece {tvhiih was properly a iranjlation of
it") done by a very ingenious Gentleman, who, tho* capable if
carrying the matter a great deal farther, rcds contented to foU
low his Original; adding only to it the computttion of the Ad-
vantage of tbe Setter in ibe Play called Hazard, and fome
few things more. As for the French Bock, I bad ran it ovtr
but eurjorifyy by reafon I had obftrved that the Author chiefy
iiffied on the Methcd (f Huygens, trhich I trot abfolutely re-
A folvtd

fi PREFACE

folved to rr/eclj ^s not fe>'*f:ino to me t:^ be the geattifte drid
nxturgl n\i/ of coining at the Soluttan of i^rohlefns tf this kind,
ftomiefy h id I allowed my felf s httle T:?oye time to con/ider
it^ I had certainly done the Jujlicc to its Author^ to have o^va*
ed that he had not only illti(frjtcd HuygensV Method hy a gr^at
variety of tveU cbojen Examples, hut that he had t.lled to it
fever al curious thi/jgs of h:s t?»v,t hvcn'inn.

Thif I have not foUoived Mr. Tfiiygtns in h'i Method cf
Solution J ^tis mth very great fleapire that I ack/;otrledge the Obli^
gations I have to him\ his Book having Settled in rnj Mind
the firfi Notions of this Doifriney and taught nte to argue about
it with certainty.

I had faid in my Specimen^ that Mr. Huygens was the
firfi who had Publifhed the Rules of this Calculation^ intending
thereby to do juflice to that great Man ; but what I then f^d
was mifinttrpreted^ as if I had depvted to wrong fome Perfons
who had confidered this matter lefore him^ and a faffage was
cited againfl me out of Huygens jr Preface^ in which he fatth^
Sciendum vcro quod jam pridem, inter Prsflantiflimos toc&
Gallic Gcometras, Calculus hie fuerit agitatus; ne quisin-
dcbitam mihi prims Inventionis gioriam hac m re
tribuat. But what follows immediateh rff^f had it bee^
mindedj might have cleared me from any Sufptcion of injufiice.
The words art tbefe Cxcerum illi difficillimis quibufque
Quxftionibus fe invicem exercere Soliti^ methodum Suam
quifque occulcam retinuerc, adco ut a primis elemenris
hanc materiam evolvere mihi neccfle fuerit. By which it
affearSy that tht? Mr. Huygens was not the firft who hoi
applied himfelf to thofe forts of QueJUons^ he was rteverthelefs
the firfi who had publifhed Rulesjhr their Solution i which is
mU that I afflrmei.

Since tie printing of my Specimen^ Mr. de Monmorr, ^«.
thTr of the Analyfe des jcux de Hazard, Publifhed a Second
Edition of that Book^ in which he has particularly given manjs
proofs of his pngular Genius^ and extraordinary Capacity i which
Jt^imony I give both to Truth^ and to the Friendfbif witb\
which he ispleafcd to Honour me.

Such a Trai as this is may be ufeful to fever al ends; the-
firjl of which isy that there being in the World feveral inqui*
pive Perfons^ who are defirous to know what foundation theyx

;/

P RE F AC B.

HI

gO' upoiff when they efigage in FUj^ ivhther from a moilve of
C4/.7, or hrefy Diverfion^ thtj mdjf^ bj the help of this or the
lih Tf'tlj gratifie their curiofitj^ either by takiff^ the pAins io
underjixnii tvhat is here Demonjirdtedj or etfe ntAking ufe of the
concf:4fio:ysj and taking it for granted that the Demonflrations

Am }:*^*ft.

L;v:htr ffe to be made of this Doltrim of Of.^r,as //, tha^
u t. ' *j ferve in Conjundicn with the other pa: is of the Afd*
iheMAtiiikSj at A fit introditltion to the Art of Reafoning ; it
being known by experience thai fwthing can contribute more td
the attaining of that Art^ than the confideration of a. long Train
of Confe^uences^ ^%^^^J deduced from undoubted Principles^ of
which this Book affords many Examples. To this mxf be added^
that fome of the Problems about Chance having a great Wear*
once of Simplicity^ the Mind is eafih drawn into a beliefs that
their Solution m^ be attuned by the meer Strength of natural
good Sence ; which generally proving othenpije^ and the Mijlakes
occajioned thereby being not unjrequentf ^tisprefumed that a Book
of this Kj'idj which teaches to difiinguifb Truth from what feems
JO- nearly to refemUe it^ will he looked upon as a help to good
Reafoning.

Among the feveral Mifiaies that are committed about
Chance J one of the moft common and leaft fufpeHed^ is that
which relates to Lotterys. Thue^ ff'pfoftng a Lottery wherein
the proportion of the Blanks to the Prizes is as five to onei
^iis very natural to conclude that therefore five Tickets are re^
quifite for the Chance of a Prize ; and yet it mity be proved
Demonjlrativeljfj that four Tickets are more then fufficient for
that purpoje^ which wiu be confirmed by often repeated Experience.
In the li(e mannerj fuppofing 4 Lottery wherein the proportion
of the Blanks to the Prizes is m thirty nine to Ont^ (fuch /u wst
the Lottery of 1710) it may he proved^ that in twenty eight-
Tickets^ a Prize is as like^ to be* taken as not ; which tho^ it
may jeem to contradiif the common NotionSy is neverthelefr
grounded upon infaHibU Demonftrdtion.

When the Pixy of the Rojal Oak was in ufe ^ fome Perfons
who lofl confiderably by it^ had their Lojfes chiefly occaponed by
an Argument of which thg could not perceive the Fallacy. The
Odds agatnfl any particular- Point of the BaJl were one and
Thirty to One^ which intituled the Adventurers^ in cafe thty

A .» nere

iv P R E F A C E.

fvere w^^ers^ to hsve thirty two S'lLn rK:urfj^d^ includi/ig their
oivn ; ia/fead of ivhich they ha'/i*:;i bnt erght and Twenty^ it
\vj§ tery flxin lh.it en the .S'//^': .t:rnr!i cf the dtfiAvtntagn
'/ the PLty^ they I' ft oie i!'^.;h jirt nf aP the /h'C'^-~y thy
pUfd for. B.tt the fll-jht" cf (he B .'i ..j 'ijJi.!j/;^:< r* : thct
hxd no wafo/i to €o?/:p! ii/^ \ li^tce he i-ov.v! .^^idtr: rrc t-:i^ 4///
part/cuLtr point of the BdH Jh^'/ld come /•/ /;/ two ar-i fwrniy
'throws ; of this he would vff<r to Uj a ^Vtger^ arjd a^lufiUy
Utd it when rt quired. The ftemih'^ cofJf xdidion between the
Odds of one and thirty to One^ and Trventy two jT/rows for any
Chance to come up^ fo perplexed the Adventurers^ that they
begun to think the Advantage wjs on thtsr fide ; for which
reafon they pUyd on dnd continued to loje*

The Doclrine of Chances may l/kewife be a help to cure 4
Kjnd of Super (lition^ which hits been of iong ft ending in the
WoiU^ 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 Afftfiance than that of their own reafon^ arc
fatisfied^ that the Notion of Luck is meerly Chimerical \ jet I
conceive thai the ground they have to look upon it as fuchf
may fliU be farther inforced from fome of the following Confix
itraiions.

If bj faying that d Man has had good Luck^ nothing more
was meant than that he has been generaOj a Gainer at pTay^ the
Expr^n might be allowed as very proper in a fhort way of
fpeaking : But if the Word good Luck be underjlood to fignifie
a certain predominant quality^ fo inherent in a Man^ that he
mufl win whenever be Plays^ or at leaft win oftner than l<fiy
it may be denied that there is any fuch thing in Stature.

The Afferters of Luck are very Jure from their own Expert^
ence^ that at fome times they have been very Lucky^ and that
at other times they have had a prodigious run cf ill Luck
againft themj which whiljl it continued obliged them to be very
cautious in engaging with the fortunate \ but how Chance fiould
froduce thofe extraordinary Events^ is what they cannot conceive;
They would be glad for Inftance to be Satisfied^ how they could
lofe Fifteen Games together at Piquet, ^ ill Luck had not
Jtrangely prevailed againji them. But H^ they will be pleafed to
confidtr the Rules delivered in this Book^ they wiR fee that tho^
the Odds againfl their lofing fo many times' together be very

great

' .

P 31 EF A C E.

greit^ viz. 1X7 6 j to ifjff that the PoffthtUtj of it ii not d
llrofd bj the great nefs of the OdJs^ there being O^e Chance t
3 2768 that it may JO hafPen^ from whence it follows^ that it m
Jlill p'lf:!^le to come to fafs without th Intervention of what ih
callVA iMcV.

Jj:; ^o, This Accident of lofittg Fifteen times together at PIqiic
u 4(/ >iiore to le im fated to ill Lad, than the Winning with ot
fiijglc Ticket the Highefl Prize^ in a Lottery of jxjSZ Tickets, is i
te imvutedtogood Luck, fince the Chofices in both Cafes areferfefi
equal. But if it be faid that Luck Luu been concerned in this latti
Cafe, the Anfwer iviU be eajy \ for let ttsfuffofe Luck not ixiftin
or at leafl let us fufpofe its Infiue/.*ce to be fuj fended, jet the Htgi
ejt Prize mufl faU into fome fJdnd or other ^ not bf Luck, (for <
the Hjpothefis that has been laid afide) but from the meer Necefft
cf its falling fomewhere.

Thofe who contend for Luck, msj, if tbej fleafe^ a/ledge eth
Cafes at Plaj, much more unlikelj to happen than the Winnit
or Lofing Fifteen Games together^ jet flit their Opinion will nev
receive anj Addition of Strength from fucb Suppofitions : For, ij t
Rules of Chance, a time msf be computed, in which thofe Cafes m
as prAahlj happen as not ; nay, not onlj fo, but a time maj
computed in which there maj be any proportion of Odds for tht
fo happening.

But fuppopng that Gain and Lofs werefo Jlulfuating, as oIwa
to be dfjiributed equally, whereby Luck would certainly be annihii
ted ; would it be reafonable in tbu Cafe to attribute the Events
Tlaj to Chance alone f I think, on toe contrary, ii would be qm
otherwife, for then there would be more resfon to fufpeS that for
unaccountable Fatality did Rule in it: Thus, If two Per font pi
at Crofs and Pile, and Chance alone be fupposdto be concerned
regulating the fall of the Piece^ is it probable that there fbould
an Equdity of Heads and Crojfes f It is Five to Three that infoi
times there will be an inequality ; ^tis Eleven to Five in fix, 5
to 1$ in Eight, and about ix to i in a hundred times I Whet
fore chance alone by its Nature confiitutes the Inequalities of PL
and there is no need to have recourfe to Luck to explain them*

Further^ The fame Arguments which explode the Notion of Luc
may, on the other Rde, be ufefulinfome Cafes to eflablijh a due cor,
far if on between Chance and Defign : We mry imagine Chance ju
Defign to be at it were in Competition with each other ^ for the pt

a tluui

vi PREFACE

dnltlon of feme forts of Event Sy /iniwftj cdculjde what ProiaiiUtj
there «, that thofe R^^ents [f^'^uldbe father om?jg to one thin to tne
other. To give Afitmiiiiir Irr » eofthh^Let rssfupfofethat tpyoPjtcks
of Ptqt<tXard^ ^"'^'^? f^^ ^ 'i '^ fhould be perceived th.u tW-^ rjt^
from Top io BhttO'-f, '•' e futrie Dtfpofition of the Cards /> . th
Packs \ Let //> /r:nvr.'^' lufi^if^ tbat^ Jome doubt arifing About tihs
Difpofition vf the C\.?ilSf it Jbould be qucflio/ied whether it ought
to be attributed to Chance ^ or to the Maket^s Deftgnx In this cafe
the Dolirine of Combination decides the Queflion^ fince it may be
proved by its Rules^ that there are the Odds of above 263i}o8}
Millions of Millions of Millions of Millions to One^ that the Cards
were defignedlj fet in the Order in which tbej were found.

From this lajl Conjideration we 7naj learn^ in moiij Cafes^ how
to diftinguijb the Events which are the effell of Chance^ from thofe
which are produced by Defign : Tlje very DoHrine that finds Chance
where it really is^ being able to prove by a gradual Increafe of Pro*
bahility^ till it arrive at Demonflration^ that where Vniformity^ Or*
der and Conflancy refidc^ there alfo re fide Choice and Defign.

Laftly, One of the Principal Vfes to which this Do£lriae of
Chances majf be applfdy is tne difcovering offome Truths^ which
cannot fail of pleajtng the Mind^ iy their Generality and Simpli*
city ; the Admirable Connexion of its Confequences will increafe the
Pleafure of the Difcovery ; and the feeming Paradoxes wherewith
it abounds^ will afford very great matter of Surprize and Enter*
tainment to the Inquifitive. A very remarkahle Inftance of this
nature may befeen in the prodigious Advantage which the repetition
of Odds will amount to ; Thus^ Suppopng I play with an Adver-^
far J who allows me the Odds of ^^to 40, and agrees with me to
play till 100 Stakes are won or lo(t on either fidtj on condition that
I give him an Equivalent for the Gain I am intitled to by the Ad*

vantage cf my Odds ; the Qaeftion is what Equivalent lam to give
him^ on jHppofition we play a Guinea a Stake : The Anfwer is 99 Gui»
neas and above 18 Shillings j which will feem almojl incredible^
confidering the fmalnefs of the Odds of ^t to 40. Now let the
Odds be in any Proportion given^ and let the Number of Stakes to
be played for be never fo great ^ yet one General Conclufion will in^
elude all the poffible CafeSj and the application of it to Numbers
may be wrought in left than a Minutes time.

I have explairfd^ in my Introdu£lion to tlse following Treatife^
the chief Rules on which the whole Art of Chances depends ; I hdva

done

{

?

R E F A C E vir

mighf he (as much as pojjible) of Ge/jeral Vfe. 1 fistter mj felf

cquAinud with Arithyr^*^::4 O^n.itioas^ mll^

Jane U in the flai^efi mxnntr that I co*iU think of^ to the end it
might be (as much a ^" -
; '•?/ ihofe who dre ac

bj the heh of the Introduftion aione^ h %bU to fc'-r.^e a great
yar;ety of QucjHons defending on Cha:/ce : I ivijh^ for the S/th of
fome GtntUmtn who hsroe been ftedfed to fulfcribe to the i^ri/^tf-;^
<f m) Booky that I couU eve/y rphere have been as plvn -is in rte
Incrodudion; but this washurdljpyf.[Hcable9 tlyt Invention of the
greateft fsrt of the Rules being int'trelj owing to Algebra ; jet t
have J as much as poffibUy endeavoured to deduce from the A!ge^
braical Calculation feveral praSical RttleSj tin J ruth of which
msf be depended upon^ and which maj be very ufeful to thofe tpho
have contented themfelves to learn only common Arithmetici.

^Tis for the Sake of thofe Gentlemen that I have enlarged my
frjt Dejjgn^ which was to have laid dawn fuch Precepts only as
might be fufficient to deduce the Solution ^ any difficult Problem
relating to my Subjeit : And for this reafon I have (towards the
latter end of the Book) given the Solution^ in Words at lengthy cf
fome eajjf ProbltmSf which might elfe have been made Corollaries
or Confequences of the Rules before delivered: The fingle Difficulty
which may occur from Pag« 155*/^ the end^ being only an Algebra^
ical Calculation belonging to the 49tb Problem^ to explain which
fully would have required too much room.

On this Occafionj I muft take notice to fuch of my Readers as
are well vers'^d in Vulgar Arithmetickf that it would not be J^ffi^
cult for them to make themfelves Maflers^ not only of aU the
Prailical Rules in this Book^ but alfo of more ufeful Difcoveries^
if they would take the fmall Pains of being acquainted with tltt
bare iJotation of Algebrai which might be done in the hundredth
part of the Time that is fpent in learning to read Shore-hand.

One of the Principal Methods I have made ufe of in the fol-
lowing Treatife^ has been the DoHrine of Combinations, taken in
a Sence fomewhat more extenfhe^ than as it is commonly underjlood*
The Notion of Combinations bein^ fo well fitted to the Calculation
of Chance^ that it naturally enters the Mind whenever any At*
tempt is made towards the Solution of any Problem of that kind.
It was this which led me in coarfe to the Conftderation of the Dr-
greet of Skill in the Adventurers at Play^ and I have made ufe
of it in moft parts of this Book^ as one of the Data that enter the
Quejlioni it being Jo far from perplexing the Calculation^ that on-

the-

Tiii PREFACE.

the contfdfj it is rathrr 4 f/elp ar::l »i/t OrnAmfnt to it : It u trfte^
thxt this Degree of Skill is fjnt to hi Inoxn A/ty oihtr w.ij tbM ftom
ObjervMion \ hut if the fdme Ohjtrvation cofijlAntlj recur^ *fii
firorrgtj to he frefumed thxt a rjear EfliniAtio/f of it rnxj he fnadt:
However^ to make the Calculation trtore prccife^ a/td :o avoid cau^
fiMg avy needlefs Scrttfles to thofe who love Geometric J Exailttefs^
tt will be eafjj in the room of the Word Skill, to ftthftttutest Greater
or Lefs Proportion of Chances among the Adventurers^ fo as each
of them tKay he faid to have a certain Numier of Chances to win
one fingle Game.

The General Theorem invented h) Sir Ifaac Newton, /or raifing
a Binomial to any Power given, facilitates infinitely the Method of
Combinations^ reprefenttng in one View the Combination of all the
Chances, that can happen in any given t^ umber of Times. ^Tii
by the help of that Theorem, joined with fome other Methods, that
I have been able to find fraHical Rules for the fotving a great
Variety of iifficnlt Quejlions, and to reduce the Difficulty to a fin*
gle Arithmetical Multiplication, whereof fever al Injlances may he
jeen in the 21ft Page of this Book.

Another Method I have made ufe of is that of Infinite Series,

%vhicb in many cafes wiB folve the Problems of Chance more natu*

rally than Combinations. To give the Reader a Notion of tbis^

tve may fupPofe two Men at Play throwing a Die, each in their

Turns, and that be be to he reputed the Winner who fhaU Jlrft throw

an Ace : It is plain, that the Solution of thu Problem cannot fo

properly be reduced to Combinations, which ferve chiefly to Jeter*

mine the proportion of Chances between tie Gameflers, without

any regard to the Priority of Play. '^Tis convenient therefore to

have recourfe to fome other Method, fuch as the foBowing. Lit tse

fuppofe that the fir ft Man^ heing willing to Compound with his Ad*

verjary for the Advantage he is intitled to from bis firfi Throw,

fhould ask him what Conftderation be would allow to yield it to bins \

it may naturally be fuppofed that the Anfwer would he one Sixth

part of the Stake, there being hut Five to One againfi him, and

that this Allowance would be thought a jufl Equivalent for yielding

his Tljrow: Let us likewife fuppofe the Second Man to require in

his Turn to have one Sixth part d the remaining Stake for the

Conftderation of his Throw ; which being granted, and the firfl Maifs

Right returning in courfe, he may claim again one Sixth part of the

-Remainder, and fo on alternately, till the whole Stake he exhaajted:

But

PREFACE. i«

Bat tha not leing to he done till after 4n iafinite nmnher M Shgret
he thtti taken on both Sides^ it behnos to the AJethd ff In ftflitt
Scries to ajji'^n to each Mm what proportion ef the Stake he ought
:o take at fir ft ^ fo at to ^w/wer (Xsfllj that ftrttiioM OivtfioH ^
the Stikexn'xnfivXrum; bf means of which it triB be foanJ^
that the Stake ought to Ic Divided betneen the contending Par*
ties into two Parts ^ rtfpe6fivelj proportional to the two If umbers 6
and y, Bj the like Method it would he found thai if there were
Three or more Adventurers playing on the conditions above defcri*
bed^ each Man^ according to the Situation he is in with refpeS to
Priority of Play^ might take as his Due fuch part of the St'ake^ m
is expref/ile by the corresponding Term of the proportion of 6 to ^^
continued tofo many Terms its there are Gamefiers\ whsch in the
cafe of Three Ga^nefierSj for Inflance^ would be the If umbers 6, f
and 4^, or their Proportionals jtf, 30, endtf.

Another Advantage of the Method of Infinite Series «r, that
every Term of the Series includes fome particular Circumftanc^
wherein the Gameflersmay be founds which the other Methods do not i
and that a few of its Steps arefufflcient to difcover the Lam of its
Procefs. The only Difficulty which attends this Method^ being thgt
of Summing up Jo many cf its Terms as are requifite for the So*
lution of ski Problem propofed: But it will he found by experience^
that in the Series refulting from the conftderatiomf mofl Cafes re*
lating to chance^ the Terms of it will either con/litute d Geometric
Prcgrejfion^ which by the K/sown Methods is esfily Summabki or
elfe fome other fort of Progreffion^ whofe nature confifls in thi%^
that every Term of tt has to a determinate number of the prece^
ding Terms J each being taken in order ^ fome conftant relation i ist
which cafe I have contrived fome eafieTheoremSy not only for finding
the Law cf that relation^ but alfo for finding the Sums requhredi
as may befeen in fever al places of this Book, but particularfyfrom
page 1x7 to page 134. / hope the Reader will excufe my not
giving the Demonft rations of fome few things relating to thi$
Subjeltf efpeciaHy of the two Theorems contained in page 134 and
I54y and of the Method of . Approximation contained in pagf
149 and I f O ; whereby the Duration of Play is eafily determi^
ned with the help of a Table 0/ Natural Sines: Thofe Demon fira^
tions are omitted purpofely to give an occajpon to the Reader to ex*
ercifebisownJmenuity. Jntkemean Time^ I have depofited tbeno
with the Royal Sociecyi in order to be Pablifbed when itfbaS bo
thought requifite^ b A

% PREFACE.

A Third AivMttff ofthf. Method ofjnfimte Series i, thai the
SolatloHS derived from it have /t certains Generality dnd FU^dncy^
fphl:ij j'c^rce anj other Method Cd)i tit^nin to\ thofe Methods hing
dlivays perplexed with y^rioM unhw-j/i Q^nntities^ and ihe SoIk^'
tiom ovtained by them termr/yuin^ co^nuonfy in partictiltw Cafes*

There are other Sorts of Series, which tho^ not properly i/^finitef
yet are called Series^ from the Regularity of the Terms whereof they
are compofed \ thofe Terms foDowitig one another with a certainr
uniformity^ which is always to be defined. Of this nature is the Theo»
stem given by Sir Ifaac Newton, in the Fifth Lemma of the third
Book of his Principles, for drawing a Curve through any given num*
her of Points \ of which the Demonflration^ asweHasof other things
belonging to the fame Subjeil^ maybe deduced from the firfi Propoptiom
efhis Methodus Diffarcnmlisj printed with fome other of his Traffs^
iy the csre of my Intimate Friend^ and very skilful Mathematician^
Mr. W. Jones. The abovementioned Theorem being very ufeful
in Summing up any number of Terms whofe lafl Differences are equal
(Such as are the Numbers called Triangular^ Pyramidal^ &c. tho
Squares^ the Cubes ^ or other Powers of tJumbers in Arithmetic Pr^
grrffion) J have (hewn in man) places oj this Book how it might be
applicable to thefe Cafes. I hope it will not be taken amifs that I have
afcribed the Invention of it to its froper Author^ tho V/> pofftble
fome Perfons may have found fomething like it by their own Sagacity.

After having dwelt fome time tipon Various (^ftions depending o»
the general Principle of Combinations^ as lat3^down in my Intro*
dufliont and upon fotne others depending on the Method of Infinite
Series^ I proceed to treat of the Methoaof Combinations properly fo
called J which I /hew to be eafily deducible from that more general Prin^
ciple which hath been before explained : Where it may be obferveslf
that altbo^ the Cafes it is apptyed to are particular ^ jet the toay of
reafoning^and the conjequences derived from it^ areffneral ; that Me^
thod of Arguing about generals by particular Examples^ being i>
my opinion very convenient for eafing the Reader* s Imaginatioss.

Having explained the common Rules of Combination^ and givers
4 Theorem which msy be of ufefor the Solution of fome Proilemt
relating to that Sukieity I lay down a new Theorem^ which*Js properly is>
contrail ion of the former^ whereby fever al Qjiefiions of Charge are
refolved with wonderful eafe^ tho^ the Solution might feem ^firfij^ht
to be of infuperabU dsficutty^

PREFACE xi

// is ij the Help of that Theorem fo co/ftraifeJ^ thdt I hxvt item
ahle to give a co}?ifkAt Solution of the Problems i^Pharaoo 4fid
liailcie, which was ^jn)€r done tin notp I lov:nthdtfomegreaiM4*
shefnUici^ns have before }?$& taken the Pains of calculating the Ad*
vantage of the Banker, in an) circumpance either of Cards remain^
ing in his Han As ^ or of anj number of times that the Card of the
Ponte is contained in the Stock : But fliU the curiofity of the jfnqui*
Ji:ive remained unfatispedf The Chief Queflion^ and ty much the
mofl difficulty concerning Pharaon or Bauete, hcing what it h that
the Banker gits per Cent of all the Money adventured ai thofe
Games^ which notp I can certainly anfiver is very near Thrie per
^ '*' "^ '' ' ^ ^i&xtj4sm£)

^ is calculated
. . . , . ^Algebra,

whereby fjme Queftions relating to Combinations arefolvedfyfi ea/y
a TrocejSj that their Jolution is made infome nseafure an intme*
diate conjequence of the Method of Notation. I wiB not pretend t^
fay that this new Altvebra is abfoluttly necejfary to the Solving of
thofe Que fi ions which I make to depend on it^ pnce it af fears by Mr.
De Monmort's Botk^ that both he and Mr. Nicholas Bernoully
havefolvedy by another Method^ many of the cafes therein frofofeai
But I bofe IfbaH not be thought guilty of too much Confidence^ if
I affure the Reader^ that the Method I have followed has a degree
of Simplicity f not to fay of Generality^ which wtB hardly be attained
ty anj other Steps than ly thofe I have totem.

The r^th TrolUm^ pfopofed to me^ aynor^ fome others^ by
the Honourable Mr. Francis Robartes, Ibadjolvedinmy TraS De(
menfura Sortis; // relates^ as weS as the r/^th andiffthj to the
Method of Combinations^ and is made to depend on the fame Prsnci--
pie ; When I began for the frfi time to attempt its Solution^ I had
nothing elfe to guide me but the common Rules if Combinations^-
fucb as they had been delivered by Df. Wallis ahd others ; which
when I endeavoured to apply j I was Stirpfized ^o find that my *
calculation fweUed by degrees to an Intolerable bulk: tor this re afore
I was forced to turn my Views Another wsy^ and to try wheshtt
the foiution I was feeking for might not be deduced from jome eaft&
considerations I whereupon I happily fei sipon the Method I have been
mentionin^^ which as it led me to a very great SimPlitity in the
Solution f fo I look upon it to be an Impravemene maJii to tbiMe^
tbod <f CombinatioitSm

Tie

xU PREFACE.

The ^Oth Problem U the reverse of tlie frutding\ It C4inUim
A very remarkable Method of Solutiofty the Jrttpce of n^hicij r<» v-
fi^s in chM2ing an Ar'nhmetic Progrefjlon ot t^umbers intii .<
Geometric one^ this bei/jg alwajs to he done when t! c t^umhen ♦/v
Urge^ and their Intervals fmall. 1 freelf acknowledge that I h^ve
been indebted long ago for this ufefal Idea^ to mj much ref petted
Friend^ Tl)at ExceBet^t fl^ldthematicidn Dotlor H alley. Secretary
to the Royal Society, whom I have feen practice the thing on am
.other occafion: For this and other Infirutlive Notions readily
imparted to me^ during an uninterrupted Friendfhip (f five and
Iwentj jears^ I return him nty very hearty Thanks.

The 3 %d Problem^ having in it a Mixture of the two Methods
of Combinations and Infinite Series^ may he propofed for a patter m
. of Solution^ infome of the mqft difficult cifes that may occurr in the
SubjeS of Chance^ and on this occafion I m:tft do that "Ju^tce ta
Mr. Nicholas Bernoully, the iVortfjy Profejfour (f Mathematics
at Padua, to own he hadfent me the Solution of this Problem he*
fore mine was Pubhfbed; which I had nofooner received^ hut/com*
municated it to the Royal Society, and reprefented it as a Perfor*
mance highly to he commended: Whereupon the Society order'* d that
his Solution fbould he Printed^ which was accordingfy done fame
//xvei^fr//r/i&ePhilofophicalTraafaftiofi8, UunA. 341. where
mine w/u alfo inferted.

Jhc Problems which follow relate chieflf to the Duration ofPlgy^
or to the Method of determining what number of Games may pro-
bably he played out by two AdverfarteSj before a certain number of
Stakes agreed on between them be won or lofi on either fide. This
SubjeH affording a very great Variety of Curious Quejlions^ of which
every one has a degree cf Difficulty peculiar to it felf 1 thought it
neceffary to divide it into fever al diftinU Problems^ and to iSse^
Jlraie their Solution with proper Examples.

Xh(f thefe Queflions may at firjl fight feem to have a very great
Jegree (f difficulty^ yet I have fome rerfon to believe J that the Steps
J have taken to come as^ their Solution^ wiU eafily be foUoweJ by
thofe who hjove a competent skill in Algebra, and that toe chief Me*
thod of proceeding therein will be under flood by thofe who are barely
acquainted with the Elements of that Art.

When J firfi began to attempt the general Solution of the

Prjoblem concerning the Duration of Play^ there was nothing ex-

jant that could give mi any light into that SulgeSi for altb^Mr.

de

PREFACE. xiii

de Monmorti in the firjl Edition of hii Book^ giveitheSohtiim
of this Proiler/tf as limited to three Strikes to he rtou or lofi^ dnd
farther litrited tj the Sffppo/ition of an Eqajlitj .y .Skill ietween
the Adventurers ; jet be hrji^ig ^iven m Oeny^iftrMion cf kit
'J^fr^tixvj and the DfnfO/{flr.ifio:f when Ifccrjsr^i f^n/sg vf very lit*
i\' :fjc toivards ohtaifting the general Stln^tn,, oj the Piublem^ I
ii'.jj forced to try what my own Enquiry xvo;dd lead me to^ which
haviiig been Attended tvith Succefs^ the rtfuh cf what I found itas
afterwards fublifhcd in my Specimen Before mentioned.

JB the Problems which in my Specimen related to the Dara*
tion of Play^ have teen kept entire in the following Treatije ; .but
the Method of Solution has received fome improvements by the
new Difcoveries I have made concerning the Nature of thcfe Series
ivhicb refult from the Confideration of the Subjeif ; however^ the
Principles of that Method having been laid down in my Specimen
/ had nothing now to da, but to draw the Confiquences that were
naturally deducible from them*

Mr. de Monmort, and Mr. Nicholas BernouHy, have each of
them feparately given the Solution of my xxxix/^ Problem, in a-
Method differing from mine, as may be feen in Mr. de Monmort^i
fecond Edition of his Book. Their Solutions, which in the main-
agree together, and vary little more than in the farm of Expreffion,
are extreamfy beautiful; for which reafon I thought the Reader
would be well pleafed to fie their Method explained by me, in fuch a
manner as might be apprehended by thofe who are not Jo well verfedin*
the nature of Symbols: In which matter I have taien fome Pains,
thereby to tefiify to the World the jufi Value I have for their
Performance.

The 43d Problem having been propofed to me ly Mr. Thomas
Woodcock, a Gentleman whom I infinitely refpeS, I attempted its
Solution with a very great defire of obtaining it; and having had
the good Fortune to fucceed in it, i returned him the Solution a few
Days rfter he woe pleafed to propofe it. This Problem is in my
Opinion one of the mofi curious that tan he propofd on this Subje^ \
its Solution containing the Method of determining, not only that
Advantage which refult s from a Superiority of Chance, in a Play
confined to a certain number of Stakes to he won or lofi by either
Party, but alfo that which may refult from an uteequality of Stakes ;
and even compares thofe two Advantages together, when the Odds cf
Chance beittg on one fide, the Odds if Money are on the other.
' c Before

xW PREFACE.

Btfcre I tuskt m t^d ef this Difaurji, I thliA nrf ftlfpbligtd
to tdkt t^ciice, that fomt Teitrs afttr wij Specimen m.{t frintitlf
thtre <:*me out a TrdSt upon th Suhjtll if ilhance^ Seix^ * ToS'
bumntti Work if Mr. James liunioully, wherein the Aulhtr bm
fhtiBii If gredt diil of fiii/l a»J fudgmenl^ and PfrfeiJlj nafnerti

the Chtrs0er atiJ grt,it Refutation be htihfojaflu obtained. Inifh
I xttre (i^ahU of currjing on » Projeii he had Begun, of a^iPlyi
the Doilrine of Chaneei to Oeconomical &ad Political 'OfeSy

whith I h&vt heen invited, together with Mr de Moiimort, hy Mr,
Nicholas Bernoully .- Jhetrtilj thank that Gentleman for the good

into better HandSy tvijhing that eith*r he himfelf would profetutr
that Defign, he having formerlj publijhed feme fucctftfut Effajs ef
that KJndy or that hiiZ/nelef flJr.Joha Bernoully» Brother ta
the Deteafed, could he prevailed upon to be/low fotue of his Thought t
upOH it ; he being knoivn to he ferfeSlj tveR ^ualifed im aS BeJfeSi
for faeh an "Undertaking.

Due Care having been taken to avoid the Errtat of the
Ff efst we hope there are no other than thefe two»

Pag. jy. Lin. 3f. for » — 3 read » — i.
Pag. j(S. Lin. x. for « — 3 read » — 1.

^Mism

•^j'ipS, V9'

Tfae

The DOCTRINE

CHANCES:

INTRODUCTIOR

|HE Probability of an Erent is greater, or
kfs, according to the number of Chances
b/ which it may Happen, compar*d with
the number of all the Chances, by which
it may either Happen or Fail.
Thus, If an Event nas 3 Chances to Hap-
pen, and X to Fail ; the Probability of in
Happening may be eAimaced to be ^ and the Frooabilitjr
or its Failing Jl,

Therefore, if the Probability of Happening and Failing
lire added together, the Sum will always be equal to
Unit;-.

B If

2 ' ••• The Doctrine of Chamces-

'••• if. ihc'.Erobabilitics of Happening ^nd Failing arc unequal^
there is what is commonly call'd Odds for, or agiinft, the
Happening or Failing; which Odds arc proportional to the
number ot Cliances lor Happening or Failing.

The Expectation of obiahiing any Thing, is cftimatcd by
the Value of that Thing multiplied by the Probability of
obtaining it.

Thus, Suppodng that J and B Play together ; that ji has
depoficed 5 /» and B 3 /; that the number of Chances which
J has to win is 4, and that the number of Clianccs which
B has to win is x : Since the whole Sum depoGted is 8, and
that the Probability which A has of getting it^is ^; it fol.

lows, that the Expe£lation of J upon the whole Sum depo-
fited will be 4->c-7-= 5 4- 9 ^nd tor the &me reafon. the Ex-
peftation of B will be ix 4- = x — .

* 1 o %

The Risk of loHng any Thing, is eftimated by the Value
of that Thing multiplied by the Probability of loflng it.

If from the refpective Expeflations, which the Gamefters
have upon the whole Sum depofited ; the particular Sums they
depofity that is their own Stakes, be fubtrafted, there will
remain the Gain, if the difference is pofitive, or the Lofs, if
the difference is negative.

Thus, If from 5-L the Expc6btion of Jy 5 which is his

own Stake be fubtraded, there will remain -2- for his Gain ;
likewife if from i-L the Expeflation of B, 3 which is his
own Stake be fubtraSed, there will remain —* ^ ^^^ ^^ Gain,
or JL for his Lods.

Again, If from the refpeflive Expeflations, which either
Gamcfler has upon the Sum depofited by his Adverfary, the
Risk of lofing what he himfelf depofits, oe fubtra£ted, there
will likewife remain his Gain or Lofs.

Thus, In the preceding Cafe, the Stake of B being 3, and
the Probability which A has of winning it being -i-, the

£xpc£lation of A upon that Stake is -i-x— = ^ = x.

Moreover the Stake of A being y, and the Probability of
lofing it being ^, the Risk which A runs of lofing his

own Stake is ^x-r- =-t = ir^. Therefore, if from the

Ex-

f

The Doctrine 0/ Chances. ^

Expe£tatioQ x» the Risk 1 4-, be fubtrafted, there will
main

3Q x» the Risk i -|-, be fubtrafted, there will tt*

, as bcforey for the Gain of A ; and by the fame way

oF arguing, the f^fs of B will be found to be -i-

U. B. Tho^ the Gain of one is the Lofs of the other, yet
it will te convenient to look for thcni fevcrally, that one
Opciadon may be a Proof of the other.

If there is a certain number of Chances by which the
podcflion of A Sum can be fccurM ; and alfo a certain num-
ber of Chances by wliich it may be loll ; that Sun^ may be
Infured for that part of it, which (hall be to the whole, as
the number of Ounces there is to lofe it> to the number of
all the Chances.

Thus, If there are 19 Cliances to fecure the pofledion of
1000 /, and I Chance to lofe it, the Infurance Money nuy
be found by this Proportion.

As xo is to I, fo is 1000 to 50 ; therefore $0 is the Sum
that ought to be given, in this Cafe, to Infure 1000.

If two Events have no dependence on each other, fo that f

be the number of Chances by which the firft may Happen^

and q the number of Chances by which it may Fail| and

likewife that r be the number ot Chances by which the le*

cond may Happen, and s the number of Chances by whidi

it may Fail : Multiply f + } by r -\- j, and the Produ£tr

; fr + jr'\'ps+qs will contain all the Chances, by which the

1 Happening, or Failing of the Events may be varied amongft

] one another.

i Therefore, If A and B Play together, on condition that

I if both Events Happen, A (hall win, and B lofe ; the Odds

that A (hall be the winner, are as fr to qr+ps-^-js; for the*
I only Term in which both f and r occur is pr ; therefore the

j Probability of jfs winning is ^+^^,,+^ ' ;

But if A holds that either one or the other will Happen ;.
the Odds of A^s winning are as pr + qr+ps to jsi for fome
of the Chances that are favourable to Af occur in every one
of the Terms ^r, jr^ ps.

Again, If A holds that the Hrft will Happen, and the fe-
cood Fail; the Odds are as psto pr^qr^ js.

From?

-^..

r*"^*''*^ -•••■ #a^^w^i«p«>WiiM^iWiMiaMwiw<i^>i«WI*iM'-<n*>^^^>«"^> ■■"^■'^■■■•^••^^■^^■v^^^^M

4 T^he Doctrine 0/ Chances.

From what has been fiid, it follows, chat if a Fra£lion ex«
prelTcs the Probability of an Event, and another Fraflion the
Probability of another £vent, and thofe two Events are in-
dependent ; the Probability that both thofe Events will H^
pen, will be the Product of thofe two Fra^lions.

Thus, Suppofe I have two Wagers depending, in the firft of
which I have 3 to 2 the bed of the Lay, and in the fecond
7 to 4, what is the Probability I win both Wagers ?

The Probability of winning the fird is X, that is the nuoi-

ber of Chances I have to win, divided by the number of all
the Chances ; the Probability of winning the fecond is -2.;

Therefore multiplying thefe two Fraftions together, the Pro-
duct will be -^ which is the Probability of winning both

Wagers. Now this Fradion being fubtraCled from i^ the
remainder is JL, which is the Probability I do not win both

Wagers : Therefore the Odds againH: me are 34 to x j.

%"" If I would know what the Probability is of winning
the (irft, and lodngthe fecond, I argue thus; The Probability
of winning the firft is X, the Probability of lofing the fe-
cond is -j^: Therefore multiplying -i by -^, the Produ£l-i2»

will be the Probability of niv winning the firft, and lofing the
fecond; which being lubtraaed from i, there will remain -ii.

which is the Probability I do not win the firft, and at the
fame time lofe the fecond.

3* If I would know what the Probability is of winning
the fecond, and at the fame time lofing the firft; I fay thus,
the Probability of winning the fecond is JU, the Probability
of lofing the firft is X. Therefore multiplying thefe two
Fraftions together, the Product ii. is the Probability I win
the fecond, and alfo lofe the firft.

4* If I would know what the Probability is of lofing both
Wagers ; I fay, the Probability of lofing the firft is JL, and

the Probability of lofing the fecond ^ ; therefore the Proba-
bility of lofing them both is -1-, which being fubtra£led
from I, there remains ^; therefore the Odds againft lofing
both Wagers is 47 to i.

This

jffijliffmafnmmm i ^ll . ' i i >«ii . iwiyi(iiy^>tt>'^pc^wr!'y>*f*P'"P"S^^

mfmi^m'm'^tft*'^^^

The Doctrine 0/ Chancer %

This way of realbumg is plain, and ts of veiy great extenCp
i}cing applicable to the Happcningor FailingoFas many Events
as may fall under confidcration. Thus, if I would know what
the Probability is of miflTing an Ace a times together with a
common Die, I confidcr the miffing of the Ace 4 times, as the
Failing of 4 different Eventsi; now the Prcbability of miffing
the firit is -^, the Probability of miffing the fecood is alfo JL, the

thii d ^ the fourth -|> ; therefore the Probability of miffing the

Ace 4 times together is -i. x-i x -f- x -1-=-^ J which being

fubtrafled from i, there will remain -^ for the Probability of

throwing it once or oftner in 4 times; therefore the Odds o^
throwing an Ace in 4 times is 671 to 6x%.

But if the flinging of an Ace was undertaken in ) ttmesi,
'tis plain that the Probabilitjr of miffing it 3 times would
be -|-'X^ x-|-=iiTr' ^^^^ being fubtraded from r, there wilt
remain^ for the Probability of throwing it once or oftner

in 3 times ; therefore the Odds againfl throwing it in 3 timet
.are ix; to 91.

Again, fuppofe we woifd know the Probability of thrown
ing an Ace once in 4 -throws and no more: Since the Proba*
bifity of throwing it the iirfl time is ^ and the Probability

of miffing it the other three times is ^x-J-x-j-t it follows
that the Probability of throwing it die firft time, and mill
(ing it afterwards three times fucceffivdy, is ^x-|-x-|-x-^
s J^; but becaufe it is poffible to hie it in every throw as weD

its the firfl, it follows, that the Probability of throwing it once,
in 4 throws, and miffing the other three times, is 12^ sJ^

M^hich being fubtraded from i, there wifi remain <-^ for tb^

Probability of not throwing it once and no more in 4 times;
therefore if one undertakes to throw an Ace once and no more
in 4 times, he has 500 to 796 the worft of the Lay, or ; to 8
very near.

Suppofc two Events are fuch, that one of them has twice as
many Chances to come up as the other ; what is the Probable
lity that the Event which has the greater number of Chances
to come up, does not Hapjien twice before the other Happens
once } which is the Cafe of flinging Seven withtwo Dice, be*

C fore

I .. J IB • ■■■ I ■ I .1. 1 ii um <i ^i^ n i n i . i I law m w i.\ II r ■i|ii i » ■I I ! ,"" ""'^ ' ' " ' '^**?!ff!l

6 the Doctrine o/Chances.

fore four once ; fiiice the number of Chances are as x to i»
the Probability of the firft Happening before the fecond is JL,

but the Probability of its Happening twice before it, is but
« X— or ^ ; therefore 'tis j to 4, Seven does not come up

twice, before Four once.

But if it was demanded what mufl: be the proportion of
the Facilities of the coming up of two Events, to make that
which has the mod Chances, to come up twice, before the
other comes up once ; the anfwer is i x to ; very near, (and
this proportion may be determined yet with greater exaft«
nefs) for if the proportion of the Chances is ix to 5, it follows,
that the Probabtlicy of throwing the firft before the fecond
is ^, and the Probability of throwing it twice, isJlx-^ or
^ ; therefore the Probability of not doing it is ^^tj^ there*

fore the Odds againft it are as 14^ to 144, which comes
very near a proportion of Equality*

What we have faid hitherto concerning two or more E-
vents, relates only to thofe which have no dependency on each
other ; as for thofe that have a dependency, the manner of
arguing about them will be a little alterM : But to know in
what the nature of this dependency confiftsy I ihall propofe
the two following eafy Problems.

Suppofe there is a heap of x 3 Cards of one colour, and a-
nother heap of 13 Cards of another colour; what is the Pro-
bability, that taking one Card at a venture out of each heap^
I Ihall take out the two Aces ?.

The Probability of taking the Ace out of the firft heap is ^ ^

the Probability of taking the Ace out of the fecond is alfo JL .

therefore the Probability of taking out both Aces is JL ^^ jl.

ss -X., which being fubt^Qed from i, there will remam

therefore the Odds againft me are 168 to x.

But fuppofe that out of one (ingle heap of 1 3 Cards of one co-
lour, I (hould undertake to take out, firft the Ace, fecondly the
Two s tho' the Probability of taking out the Ace be JL., and

the Probability of taking out the Two be likewife JL, yet the

Ace being fuppofed as taken out already, there will remain
only I X Cards in the heap^ which will make the ProbabiUty

of

'y"''''''^'''*'''^^**^*^?*^'*— — '*^— '^^^^•■''•*iw^'^"*»**»^i*««p»w''^^**fip^"*«^"»*^w'wi^^»^f»i^

The Doctrine c/Chance^ Y

of talcing out the Two to be ^, therefore the Probability of

taking out the Ace, and then the Two, will be -^x — • And

upon this way of reafoning may the whole Doflrine of Com-
bmacions be grounded, as will be (hewn in its place.

. It is plain that in this lail Qiicftion^ the two Events propo^
fed have on each other a dependency of Oxler, which depen-
dency confifts in this, that one of tne Events being fuppofed
as having Happened, the Probability of the other's Happen-
ing is thereby akerM ; whereas in the firft Queftion, the tak-
ingof the Ace out of the firft heap does not alter the Proba*
bility of taking the Ace out of the fecond ; therefore the In-
dependency of Events confifts in this, that the Happening of
one does not alter the degree of Probability of the others
Happening.

We have feen already how to determine the Probability
of the Happening of as many Events as may be affigned, and
the Failing of as many others as may be afligned iikewife,
when thofe Events are independent : we have feen alfo how
to determine the Happening of two Events^ or as many as
may be afligned wheia ^they are Dependent .

But how to determitie in the cafe of Events dependent, the
Happening of^ as many as may be afligned, And at .the fame
time the Failing of as many as may likewife be afilgned^ is a
difquifition of a higher nature, and will be fliewn afterwards.

If the Events in queftion are in* in number, and are fuch
as have the fame number 4 of Chances by which they may
Happen, and likewife the fame number t of Chances by
^bich they may Fail, raife dJ^b to the Power n.

And if ^ and B play together, on condition that if either
one or more of the Events in queftion do Happen, A ihall winy

and B lofei the Probability of ^/ winning will be-^^=pi.*

and that of E*s winning will be ^L, s for when dJ^h is

actually raifed to the Power ;f , the only Term in which d does
not occur is the la ft ^'; therefore all the Terms but the laft
are favourable to ^.

Thus, if » = } ; raifing 4+i to the Cube 4» + y^b^itbh^k^.
all the Terms but b^ will be favourable to A } and tberelbrc

the

, . . ,■■■,. ■.dp — . , ...,,.,■■■■■, , ,, n III 11 1 ' . "' '" '

{ TBe Doctrine of Chances.

the Probability of J*j winning will be ' — ^aT — °f
- "t ^ !r'* - ; and the Probabiliry of B'j winning will be =x|iT*
But if J and S piay, on condition that if either two or
more of the Events in quellion do Happen, A Oiall win t but
in cafe one only Happens or none> B ftull win; the Probabi-

lity of J*s winning will be ^—

rr?-

-; for the on-

ly two Terms in which mm does not occur are the two la(^
■wSfMMt'"* andt** And fo of.tiieTeft.

The

The Doctrine ^Chances.

The Solution of fever al forts of Vrohkmsi
deduced from the Kales laid dovon in tbc
Introdu^ion,

PROBLEM I.

i^TT^ in 8 throws he fbAll fling Two Aces or more: IVbat
is his Probability of winmng^ or wbdi are the Odds
for or dgdinjl him f

SOLUTION.

BEcaufe there is one fmgle Chance for J^ and five againft
him, let a be made s i, and ^ =s ; ; again becauie the
number of throws is 8^ let j» be made =s 8^ and the Proba-
bility of A\ winning wiU be ^"^'l^'"/"^'"'' = ,y///,V

Therefore the Probability of his loGng will be rlTftf?!* and
the Odds againft him win be as 10156x5 to 66399i|Or as
3 to X| very near.

PROBLEM IL

TWO Mesf A d^dB fUj/ijfg d Set together^ im eaeb Game ef
the Set the number <f Chinees which A has to win is J, dnd
the number of Chances which B has to win is % : Now after fome
Games are over^ A wants 4 Games cf being uf^ and B 6 .* It is
required in this cir cum fiance to determine the Probabilities whids
either has of winning the Set.

SOLUTION.

BEcaufe A wants 4 Games of being uiy and B 6; it
follows, that the Set will be ended in 9 Games at
moft| which is the fum of the Games wanting between

D them s

•

10

The Doctrine of Chances.

them ; therefore let dJ^b be raifed to the p/A Power, viz.
/i'+9j»^+ ;6A7^i+844*i»+ I26.|J^»+ 1164**^+ Z^H'^-k^
^6a,tb'' 4- 94^' + i'; take for /^ all the Terms in which d has
4 or more dimcnfions, and for B all the Terms in which t
hiJS 6 or moicdimcnfions; and the Proportion of the Odds will
be, as 4»+9j'*+ 3647^^4. 844''^J 4 iifij^i* + ii64*i% to
84 J '^* + ^6dd'' + 9ji' + ^^ Let now 4 be expounded by 3,
and ^ by 1 ; and the Odds that A wins the S;^c will be found
as 1759077 to 1P4048, or very near as 9 to i.

And generally, fuppofing that p and q are the number of
Games refpeclively wanting; raifc a-\-b to the Power ^+9 — i,
then take for A^ a number of Terms equal to q^ and for £, «
number of Terms equal to f.

REMARKS.

1. In this Problem, if inftead of fuppofing that the Chances
which the Gamcfters have each time to get a Game are ia
the proportion of a to ^, we fuppofe the Skill of the Game*
fters to be in that proportion, the Solution of the Problem
will be the fame : We may compare the Skill of the Game-
fters to the number of Chances they have to win. Whether
the number of Chances which ^ and B have of getting; a Game,
are in a certain Proportion, or whether their Skill be in that
proportion, is the fame thing.

2. The preceding Problem might be folved without Alge-
bra, by the bare help of the Arithmetical Principles which
we have laid down in the Introduction, but the method
will be longer : Yet for the fake of thofe who are not ac-
quainted with Algebraical computation, I ihall fee down the
Method of proceeding in like cafes.

In order to which, it is neceffary to know, that when a
Queftioii fcems fomewhat difficult, it will be ufeful to folve
at firft a Queftion of the like nature, that has a greater degree
of fimplicity than the cafe propofed in the Queftion given; the
Solution of which cafe being obtained, it will be a ftep to af«
cend to a cafe a little more compounded, till at laft the cafe
propofed may be attained ta

Therefore, to begin with the fimpleft cafe, we may fup-
pofe that A wants 1 Game of being up, and B z; and that

the

The Doctrine 0/ Chances. ii

the number of Cliances to win a Game are equal ; \n which
cafe the Odds that A will be up before B^ may be determined
as follows.

Since B \Vants x Games of being up and A i, Ms plain that
B mud beat A twice together to win; but the Probability of
his beating him once is Jl^ therefore the Probability of his

beating him twice together is J-x ^= — ; fubtraft JL from

** o 3 2 4 4

I, there remains J^ which is the Probability which A has

of winning once before B twice, therefore the Odds are as
3 to I.

By the fame way of arguing *twill be found, that if A wants
I, and £ 3i the Odds will be as 7 to i, and the Probability of
winning,-7^ and -■. refpe^lively. If A wants i, and.£ 4, the

Odds will be as 15 to i, &c.

Again, fuppofe A wants x, and B 3, what arc the Odds
that A is up before B ?

Let the whole Stake depofited between A and fi be i ;
now confider that if B wms the firft Game, B and A will
have an equality of Chances, in which ca(e tlie Expe6la«
tion of B will oe -L; but the Probability of his winning the

firft Game is ^, therefore the Expectation of B upon the

Stake, arifing from the Probability of beating A the firft tlme^
willbe-i-K i =4-

But if B lofes the firft Game, then he will want 3 of being
up, and ^ but I ; in which cafe the Expedation of B will be J^^

but the Probability of that circumftance is X, therefore the
Expedation of B arifing from the Probability of his lofingdie
firft time is -|-x4- =-v?^

Therefore the Expectation of B uDon the State i, will
be X4-JL.SS -L^ which being fubtracled from i, tliere re-
mains -LI- for the Expectation of A ; therefore the Odds are

as I z to f .
And thus proceeding gradually, it will be cafy to compofe

the following Table.

A TABLE

V m ■ pi^i wii^iMi ■ iw^^MBwiwTiwwilw^pifwja^w^T^wiw^i^iiai^pWi^p^p^pwjW^PM^r^^^^W^^r^r*"??^^!^

»-«•.*«• #■ *J»I«I !■■««■ "T 1

I -■" --,->■***- ..^ ~^^ ^ -^--

12

77j)tf Doctrine of Chances.

A TABLE of the ODDS for any number of Games

wanting, from i to 6»

51

I 2

I

7

2

2«
52

3

I

I5_

19

2

57

64

3

4

I
3-«

S

t

3»

. .'

ill

2 3

T
9

t

Id

5

^5, -1-

64 64

8>

^ 8

4

2

6

jl

11 ^

16 i6

3 4

<

7
64

8

■

n

3

S

6

•5*

P

4* 33

i2(i

latf

319
2i6

356

4 5

4

6

11

p

3)6 2J6

^8a

513

•

•

«— »

II

5 <

11

^)8 ^86

1034 1034

-

And by the fame way of proceeding, it would be eafy to
^compote other Tables, tor expreffing the Probabilities which
:rf and B have of winning the Set, when each wants a given
number of Games of being up, and when the proportion of the

Chances

The Doctrine o/Chances. 13

Chances by which each of them may get a Game Is as x to r,
or varies at pleafure ; but ttie Algebraic method explained in
this Problem anfwering all that variety ^ \is ncedlefs to infift
upon it.

PROBLEM IIL

IF A dnA B fltg with fwgU Bowis^ and fach be the Skii ef K,
that be kfiows ty Experience he can give B % Gdmes out rf i:
What is the proportion of their Skiff, or tphdt are the Odds tUdt A
Moj^ get anj one Came ajfigned f

SOLUTION.

LE T the proportion of the Odds be as js to x : Now fince
j1 can give B x Games out of ^^ therefore ^cao^ upon an
equality of Play, undertake to win 3 Games together : Let
therefore ;& *|- x be raifed to the Cube, viz. z^+ 3«<> +}«+ 1 ;
therefore the Probabilities of winning will be, as^c' to i^&e-t-
3^;+ I ; but thefe Probabilities are equal, by fuppoutionf
therefore x' ss 3 ;tx + 3^4. z, or x j&' 35 «* 4- ^zzJ^- ^zJ^ x.

and extraSing the Cube Root on both fides, z^xisz z+n
therefore z s X^ » and confequently the Odds that Jcazf

get any one Game afligned are as 3. to x> or as i to v^^— if
that is in this cafe as ;o to 13 very near*

PROBLEM IV.

JF A cam mthont advantage or (^/advantage give B x Game
out of J ; what are the Odds that A Jhall take any one Game
afftgnedf Or what is the proportion of the Chances thej have t^
win any one Game dfpgned f Or what is the proportion of their
Skill f

SOLUTION-

LE T the proportion be as zi to i ; now fince J can give
B I Game out of 3 ; therefore J can, upon an equality
of Play, undertake to get 3 Games before B gets x ; let there*

E for«

i^>*^**M**>»i^a^WM

•^r J f ! ■ *^w *■ i>'^ 11 nnj« |^r<»^^^pinif^Bmnp«^^iipii w iarr^np^iyfmmii^irm^ftm^i^^^^^^titmm

14 The Doctrine o/Chances.

fore «+ 1 be raifed to the ^th Power, whofe Index 4 is the Sum
of the Games wanting between them lefs by i ; this Power
will be c* + 4 j&' + 6c:&+ 4C+ i ; therefore the Probabili-
ties of winning the Set will be as z»^ + 4*' to 6z,z + 4^ ^- i :
But thefe Probabilities are equal by Hypothefis, fince A and B
are fuppofed to play without advantage or difadvantage ;
therefore ;&* + 4z,i :=6^* + 4«^+ i, which Equation being
folved, z will be found to be 1 .6 very near ; wherefore the pro*
portion of the Odds will be as 1 .6 to i^ or as 8 to $•

PROBLEM V.

•

To pni in how mdwj TrUls sn Event will ProBdify f/dfm
feny or how many Trials will be requifie to make it indif'
ferent to Igf on its Happening or Failing ; f^ppofing that z is the
number of Chances for its Happening in anj one 2 rialp and b the
number of Chances far its failing.

SOLUTION.

LET J? be the number of Triab;. therefore by what
has been already demonftrated ia the Introau£lioa

7Pj«-^^ =^^ or :rp* = ^y% therefore* = ug,^^ylj^,i

Moreover, let us reafTume the Equation rfji' =sx^ and
makin g a^hw i, ^, the Equation will be changed into this

7+-? *==*•• let therefore , .}- J- be raifed aftually to the

Power » by Sir Ifaae Uewton\ Theorem, and the Equation will

^ - + t+-^^+ 'V.VA%r &c.=.». In this E.
ouation, if ; = i, then will x be likewife = i; if jr be infinite^
then win x alfo be infinite Suppofe q infinity then the Equa-
tion will be reduced to ! + -*-+ -2L.^.^ SfCrs x: But

the firft part of this Equation is the number whofe Hyperbo-
KcLogarithm is Y, therefore -j =Log: x: But the Hyper-

bolic Logarithm of x is 0.693 ^^ nearly 0.7 \ Wherefore — =
o.7» and « s= 0.7^ very near. *

Thus we have afligned the very narrow limits withia
which the Ratio of x to ; is comprehended ; for it begins

with

363

..—..•u - ~.- ..

The Doctrine o/Chancest, 15

With Unity, and terminates at lad in the Ratio of 10 to 7,
very near.

But X foon Converges to the limit 0.7 q^ fo that this pro*
portion may be afllimed in all cafeSy let the Value of } be
what it will.

Some ufcs of this Propofition will appear by the following
Examples.

EXAMPLE I.

LET it be propofed to find in how many Throws one
may undertake, with an equality of Chance, to fling two
Aces with two Dice.

The number of Chances upon two Dice is 36, out of
which there is but i Chance for two Aces ; therefore the
number of Chances againft it is xi;: Multiply 3^ by a7«
and the produfl %^.€ will (hew that the number ot Tnrows *
requifite to that efiea will be between 24 and x5»

EXAMPLE n.

TO find in how many Throws of three Dic^ one may
undertake to fling three Aces;
The number of all the Chances upon 3 Dice is xKf^ out of
which there is but i Chance for 3 Aces, and xi; againft it;
Therefore let 11 5 be multiplied by a/, and the produft x 50.5 >
will fliew that the number of Chances requifite to that eflea: •
will be I f o,. or very near it.

EXAMPLE m.

I.M a Lottery whereof the number of Blanks is to the num^
ber of Prizes as 39 to i, (fuch as was the Lottery of 1710;/)
To find how many Tickets one muft take^ to make it an
equal Chance for one or more Prfzes.

Multiply 39 by 0.7, and the product X7.3 will (how that the
number of^ Tickets requifite to that eSed will be x 7, or x 8 at
moft.

Like wife, in a Lottery whereof the number of Blanks is
to (be number of Prizes; as 5 to i, multiply ; by 0.7 and the

pro-

p-y....„— , i „ , . ,,,, t , m^m»m m K m» *» ■■ m »w^»j^^—^i»»^ >■■!■■■ iw m i i — ■ ■ u ■ ■.■■ w. "' . mii i " »n*m *m

M^

i6 The Doctrine of Chances.

produS 3*5 will (Iiow, that there is more than an equaltt]|r of
Chance lo 4 Tickets for one or more PrizeS| but (ometning
Icfs than an equality in 3.

REMARK.

In a Lottery whereof the Blanks are to tlie Prizes as 39
to I , if the number of Tickets in all was but 40, this propor*
tion would be altered, for 10 Tickets would be a fufficient
number for the Expeflation of the (ingle Prisx; it being evi*
dent that the Prize may be as well among the Tickets whidi
are taken as among thofe that are left benind.

Again, if the number of Tickets was 80, ftill preferving
the proportion of 39 Blanks to z Prize, and confequcntly fup-
poflng 78 Blanks to x Prizes, this proportion would (liD be
altered : For by the Doctrine of Combinations, whereof we are
to treat afterwards, it will appear that the Probability of
uking one Prize or both in 10 Tickets would be but J||.^

and the Probability of taking none would be ^; Where/ore

the Odds agamft taking any Prize would be as 177 coa39)
or very near as 9 to 7.

And by the fameDo£lrine of Combinations it will be found
that 13 Tickets would not be quite fufficient for the ExpeQ-
ation of a Prize in this Lottery; but that 24 would rather be
two many ; fo that one might with advantage lay an evea
Wager of taking a Prize in x4 Tickets.

If the proportion of 39 to i be oftner repeated, the num-
ber of Tickers rcquifite lor a Prize will ftill increafe with that
repetition: Yet let the proportion of 39 to i be repeated ne-
ver fo many times, nay an infinite number of times, the num-
ber of Tickets requifice for a Prize will never exceed JL- of

39, that is about 17 or x8L

Therefore if the proportion of the Blanks to the Prizes be
often repeated, as it ufually is in Lotteries ; the number of Tick-
ets rcquifite for one Prize or more, will always be found by
taking ^ of the proportion of the Blanks to the Prizes.

LEM-

wpiwwuywp^-wiii II u 11 w^mrmfmmmi

363

TU Doctrine 0/ Changes. I7.

LEMMA.

TO find fjoip mAnj Chafiees there gre ttfon dtrj ntfmhr if
Dice J each of them of the fame given ntimber cf Fsas^ $^
tJjroiP dnj given number (f Points.

SOLUTION.

LE T f + 1 be the number of Points given, n tbe number
of Dice, / the number of Faces in each Die : Make
f^f^q ; y— /= r ; r— /= s \ s^f^ t &c and the num*
bcr of Chances will be

* 1 a \ ^^^

„l.xC-xi^^&c.x^

~ ^xi=ix.^=^&c. x-f x2:iix2^

+ &a

which Series ought to be continued till fome of the Faftors io
each Produfl b^ome either == o, or Negative.

N. B. So many Faflors are to be taken in each of the Pro*
dufts^x^x^&c 4.x^^x2z2 &C. as there are Unitf

in n— I.

Thus, for Example, let it be required to find how many
Chances there are for throwing Sixteen Points with Four
Dice.

But 4^5 — 336 + 6=: 1x5; therefore One Hundred and
Twenty Five is the number of Chances required

F Again,

l8 The Doctrine 0/ Chances.

Again, let it be required to find the number of Chances for
throwing feven and Twenty Points with Six Dice.

-J5.xJlx-^xJi.x^x4 =~93*04

+ -^x^xJ^x-ifx^x^x-i. =+30030

_J.x4.x-f x-^x^x^x^-x^^- „xo

Therefore 6^780 — 93104 + 30030—11x0= 1666 is the
number required.

Ltt it be required to find the number of Chances for throw*
ing Fifteen Points with Six Dice«

+ Ji.xJ2-x.i2.x^x^ =+ xoox

I a I 4 5 » ^*

But xoox — 336= 16669 which is the number requi*
red.

Coral. All the Points equally diftant from the extreams, that
is, from the lead and greateft number of Points that are upon
the Dice, have the fame number of Chances bv which tney
may be produced ; wherefore if the number ot Points given
be nearer to the greater Extream than to the lefs, let the num.
ber of Points given be fubtrafted from the fum of the Ex-
treams, and work with the remainder, and the Operation
will be fbortenM.

Thus, if it be required to find the number of Chances for
throwing X7 Points with 6 Dice : Let xj be fubtraded from
4X the ium of the Extreams 6 and 36, and the Remainder
being i $9 it niay be concluded that the number of Chances
for throwing X7 Points is the fame as for throwing 15.

Let it now be required to find in how many throws of
6 Dice one may undertake to throw 15 Points.

The number of Chances for throwing x ; Points being
1666; and the number of Chances for Failing being ^4990;
divide 44990 by 1666, the Quotient will be X7 ; Multiply X7

by

i

363

The Doctrine of Changes. i^

by 0.7, and the Produft 18.9 will (hew that the number of
throws requifite to that efTeffc is very near 19.

PROBLEM VI.

To fiai bow msfty Trials are neceffdr; to Mdke it Prctdk^
that M Event will H^ffen twice^ f^ffofiftg thai a is the
number of Chances for its Hafftning at an; one Trial, and b tbe
number of Chances for its Failing.

SOLUTION.

LE T je be the number of Trials : Therefore by what
has been already Demonftrated^ it will ap pear that
i+il* = xi*+ XAxi'""*;ormakiogj,*;: 1, jj iTjTIl*—-
X 4. i£.. Now if ; be fuppofed = i, « will be found = 3;
and if ; be fuppofed infinite, and alfo JL = x, we fhall have

j6 = Loe: x + Log: i -)-x; in which Equation the value of j&
will be found = 1.678 very nearly. Therefore the value of
X will always be between the Limits ^ j and 1.67SJ. But x
will foon converse to tbe laft of thefe Limits ; therefore if x
be not very fmalT, it may in all cafes be fuppofed = 1.678 f»
Yet if there be any Sufpicion that the VaJue of x thus
taken is too lit tle, fubftitute this Value in the Original E-^
quation 1 + -^' = x + ^ and note that Errour. If it bp

worth taking notice of, then increafe a little the value of x,
and fubftitute again this new value in the room of x in the
aforefaid Equation ; and noting the new Errour^ the value of
X may be iufficiently corre&ed by applying the Rule whicb
the Arithmeticians call double Falfe Fbutioo.

EXAMPLE L

TO find in how many throws of Three Dice one may
undertake to throw Three Aces twice.
Tbe number of all the Chances upon Three Dice being xi6^
out of which there is but one Chance for Three Aces, and
XI f againft it; Multiply X15 by 1.678^ and the Produdr
360.7 will (hew that the number of throws requifite to that
c!fe£t will be 360 or very near it»

•\%

^*ww ■■!■ I t ■ipiiii m , I ' I Ml im m'tmm'^mmvmm mt i^^w'w

W ■■■ !* . lll.w<.fifi.W

id I I ■*■

ao TJ^ Doctrine of Chakcbs.

EXAMPLE IL

TO find in bow many throws of Six Dice one may un«
dcrcake to throw Fifteen Points twice.
The number of Chances for throwing Fifteen Points is 1666,
the number of Chances for MifTing 44P90; let 44990 be di-
vided by 1 666y the Quotient will be ^^ very near : Wherefore
the proportion of Chances for Throwing and Miffing Fifteen
Points are as i to 17 rtfpedlivcly ; Multiply 17 by 1.678,
and the Prcdud a;.} will lliew that the number of throws
leguifite to that eBed will be 45 nearly*

EXAMPLE IIL

IN a Lottery whereof the number of Blanks is to the num-
ber of Prizes as j9 to i : To find how many Tickets
mull be taken, to make it as Probable that two or more Be-
nefits will be taken as not.

Multiply 39 by 1.678, and the Produft 654 will fliew
that no lefs than 6f Tickets will be requifite to that effed ;
iho' one might undertake upon an Equality of Chance to
have one at lead in 18.

PROBLEM VIL

To fiii bow ntMfiy TtIaU gre necejfgry fo mske it ProiMe
thgt M,n Event wiU Hdpfen Tlnrtt^ ¥our^ Fivt^ 8rc. times \
jupfopng that a // the number of Chances far its Hafpening im
anj one Trial j and b the number of Chances for its Failing.

SOLUTION.

LE T J( be the number of Trials requifite, then fuppo-
fing, as before a^ b:: i, ^, we (hall have the Equa-
tion 1 + -^ ' = X >c I + — + -f-x ^', in the cafeof the triple
Event; or r+31 ' =:ixi + -fL4.-E.xin + -^ xifJxV^^

In the cafe of the quadruple Event: And the Law of the con-
tinuation of thefe Equations is manifeft. Now in the firft E-
^uation if j^ be fuppofed =:= i> then will « be = f • If y be

pwipmwi ] I ji.j i hwuj.j ' k.wi

363

The Doctrine 0/ Chances. ii

fuppofed infinice or pretty large in rcfpe^ to Uoiry; then
the afortlaid Equaciony making ^•=^ will be chafed in-
to this, jc = Log. 2 ^ Log. I +T+X7i'; wherein c win

be found nearly = 2.67/. Wherefore x will always be be-
tween 5f and 1.67; q.

Likewife in the fecond Equation, if ^ be fuppofed r= i,
then will jt be = 7 ^ ; but if jt be luppofed infinite, or
pretty large in rcf pcd^ to Unity, then c = Log. % + L<^

I + ^+ ^--^M- 7--' f whence z, will be found nearly = 3.6719;

Wherefore x will be between 7f and 3.6719 f.

If thefe Equations were continued, it would be found that
the Limits of « converge continually to the proportion of
two to one.

A TABLE of the Limits.

The Value of x will always be

For a fingle Event, between if and o.69if.
For a double Event, between 3f and i«670f.
For a triple Event, between $q and ZmSjsp
For a quadruple Event, between yj and 3.671;.
For a quintuple Event, between 9; aixl 4.670^.
For a fextuple Event, between i if and 5.668;.

If the number of Events contended for, as well as the
number f be pretty large in refpe^ to Unity ; the. number
of Trials requifite for tbofe Events to Happen m times, wiU
be ^j or barely tfj.

PROBLEM VIIL

Tl:/ree Game pen A, B, C, flMj together cm tbh eondithx^
that he /hall win the Set who hasfootteft ffit a cert aim mmm^
her of Games I the frofortion of the Chances which each if them
has to get any one Game slPgnedj or^ which is the fame things the
frofortion of their Skilly being reffelUvely as a, b, C Niw of*
ter they have flayed fome time^ thn fnd tliemfelves in this circnm'
fiance^ that A wants One Game (f being uf^ B TwoGames^ andC
Three ; the whole Stake between tfjem being fuffcfed i : If 'hat is the
ExptSlation of each f

G SOLU-

* > ■'■■ ■■ —II ■ ft^^g^i' i i. uf iiiiipiiwP! i |ii— i.p—i^BIIHiTyfftlPfWW^' W.^* ' " " '' —^^ " ■■*■ " ' *** " ' l- 'w

■ ■ '

2i The Doctrine of CmanCei

SOLUTIOxM.

T M the Circumftance ihc Gameftcrs arc in, the Set will be cnd-
-^ cd in FourG.^mes at moll ; let tlicrelbre 4 + A +f beraifcd
to the fourth Power, anJ it will be a* + ^a}b + 6aM + ^ab^
+ ^ + 44'r + iiashc + 4\bU + 6sAce 4- ixabcc + 6ibcc
4- ^AC^ + » ^^cbb + ^bc^ + «♦.

The Terms a* + ^4'^ + 44 'r + 6jdcc + iiaabc + iidicc^
wherein the Dimenfions of 4 are equal to or greuer t!ian the
number of Games whicli A wants, wherein alio the Dimen-
fions of b and c are kfs than the number of Games which
B and C want re'*pe£l:ively, aic intit^ely Favourable to^^ and
are part of the Numerator of his Expe£hition.
^ In the fame manner the Terms t^ 4- ^b^c 4- Cbbcc are io-
tirely Favourable to B.

And likewife the Terms 4^' 4- 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 : Where*
fore thefe Terms are fo to be divided into their parts, that the
parts Favouring eachGamefter may be added to his ExpeSa-
rion.

Take therefore all tlie Tera)s which are common, viz.
6dabi + 44^) 4- iidbce 4- 44^^ and divide them actually
into their parts; that is l^ 6dabb into dM^ nhdb^dbbd^ bddi^
idbd^ bbdd. Out of thefe Six parts, one part only, viz.bbdd will
be found to Favour £, for 'tis only in this Term that two Df-
menflons of ^ are placed before one fingle Dimendon of 4, and
therefore the other Five parts belong to A\ let therefore 544^^
be added to the ExpeQationof A^ and 144^^ to the Expectati-
on of J?, x^ Divide 44^' into its parts, dbbb^ bdbbp bbdb^ bbbd.
Of thefe parts there are two belonging to A^ and the other
two to B\ let therefore zdb^ be added to the Expe6lation of
each. 3'' Divide 1 2Abbc into its parts ; and eight of them will
be found Favourable to J^ and four to B ; let therefore ^dbbc
be added to the Expectation of A^ and 44^^^ to the Expecta-
tion of B. 4"* Divide 44^' into its parts, three of which will
be found Favourable to Ay and one to C; Add therefore Xdc^
to the Expectation of Ay and i4c> to the Expectation of C.
Hence the Numerators of the feveral Expectations of J^ B, C,
will be relpeCti vely. i •

fimf^mmmpm^mv^f^^

363

The Doctrines/Chances: aj*

I. 4* + ±a}h + ^ifc + 6AACt + xxAihc + xxahu + %xM + iji*
a. ^4 4. 4^V + 6^r 4- 141^^ 4- i4p 4- 44^rr.

The comm on Denominator of all their Expeftations be*
iog 7+7+71*.

Therefore if 4, h^ e are in a proportion of equality^ the
OJds of winning will be refpeftively as 57, 18, 6.

If n be tlie number of all the Games th^t are wanriog,
p the number of the Gamefters, 4, fc, r, J, &c. the proportion of
the Chances which each Gamefter has refpcftively to win any
one Game afligned \ let 4 + ^ + c + it &c. be raifcd to the
Power ;» + I — f^ then proceed as before.

PROBLEM IX^

Two GamefierSf A dfid B, egcb hdviir^ i% Coutittti^ flsg^
xpith three DUe^ on condition^ thdt if 11 Points come uf^
B fidll give one Counter to Ai if 14, tifbsll give one Counter
to'R; and tbdt be fbill be tbe winner tpbo Jbd/i loonefi get dUfbe
Counters of his ddvetfsrj : IVhdt are the Probdbilities tpaS edcb of
them has of winning f

SOLUTION.

E T the number of Counters which each of them have
be =^; and let 4 and b be the number of Chances they
have refpedively for getting a Counter each caft of the Dice:.
I fay that the ProbabiUties of winning are re(be£lively as jf
to ^j or becaufe in this cafe /= 12, 4 = 27, ^=: i; as 27**
to iy*% or as 9" to 5", or as iSx^z^fiSj^St to X44r4o62$'f
which is the proportion afligned by M. Hujgem^ but without.
any Demonftration :

Or more generally.

Let^ be the number of the Counters of A^ and jr the num-
ber of^ the Counters of B ; and let the proportion of the
Chances be as 4 to ^. I fay that the proportions of the Pro-
babilities which A has to g et all rhe Counte rs of hiS adverfa-

ry will be as 4^x*' — ^' to*>X4* — 4^.
^ DEMON-

L

■ If jn w jpw w.. J -^ili f i 11 ■^■■■iii 1 i j ii j i ^iw m i ni i f f i J 1 1 I ■ II lii^^W^W^WW

24 77;e Doctrine o/ Chances.

L

f

DEMONSTRATION.

E T it be fuppofed that A has the Counters £, Fy C, H SfC
whofe number is />, and that B has the Counters If K^L
i\'c.\vhofe ajiiib;:r is j: Moreover let it be fuppofed that the
Counters are the thing playM for, and that the Value of each
of them is to the Value of the following as m to t^ in fuch a
manner that the lad Counter of J to the firft Counter of B, be
ftill in tiiat proportion. This being fuppofed, J and Bf in every
circumftance of their Play, may lay down two fuch Counters
as may be proportional to the number of Chances each has ro
et a iingle Counter ; for in the beginning of the Play J may
ay down the Counter H which is the lowed of his Counters,
and B the Counter / which is his highed ; but f/, / : : 4, t^
therefore J and B play upon equal Terms. If ji win of J?,
then J may lay down the Counter / which he has jud got of
his adverfary, and B the Counter /C; but /, /C* ^ ^j ^} there*
fore J and B dill play upon equal Terms. But if J lofe the firft
time, then J may lay down the Counter G, and B the Coun-
ter Hy which he but now got of his adverfary ; but G^ f/:z
dy hy and therefore they dill Play upon equal Terms as be*
fore. So that as long as they Play together, they Play with*
out advantage, or difadvantage, and confequently tne Pro-
babilities of winning are reciprocal to the Sums which they
cxpe£l to win, that is, are proportional to the Sums they re*
fpe£lively have before the Play begins. Whence the Proba-
bility which A has of winning all the Counters of S, is to the
Pronability which B has of winning all the Counters of A^
as the Sum of the Terms £, Fy G, H whofe number is fy
to the S um of the Term s hK^L w hofe number is ^; that

is, as 4^X4'—*^ to b^%A^--bi\ As willeafily appear if
thofe Terms which are in Geometric Pix>gre0ion are actually
fummed up by the known methods. Now the Probabili-
ties of winning are not influenced by the fuppofuion here
made, of each Counter being to the following in the pro-
portion of A to b\ and therefore when thofe Counters are
luppofed of equal Value, or rather of no Value, but fervc
only to mark the number of Stakes won or loft on either
lide, the Probabilities of winning will be the fame as we have
aiTigiied. R £.

■ ' ■■■iPi^N»"Wtp^"yrr"»iwmwniJ.p^i I I II 111 ' ■^pyyu^^U I- ii'i -n

363

The DOCTRINB c/CHANCfiS. 2$

REMARK L

IF p and j, or cither of rhcm are large numbers, 'twill be
convenient to work by Logarithms.
Thus, If A and B play a Guinea a Stake, and the num«
bcr of Chances which A has to win each fingle Stake be 43,
but the number of Chances which B has to M'in it be ao ;
and they oblige themfclves to play till fuch time as 100 Stakes
are woo and Toft.

From the Logarithm of 4J = 1.633468;
Subtract the Logarithm of 40=: 1.60x0600

birfcrencc =0.031408;

Multiply this Difference by the number of Stakes to ba
playM off, viz. 100 ; the Produft will be 3.1048500, to which
anfwers, in the Tables of Logarithms, the number 1381 ;
wherefore the Odds that A Ihall win before B are 1383
to 1.

Now in all circumftances wherein A and B venture an e-
qual Sum ; the fum of the numbers exprefling the Odds, is
to their difference, as the Money playM for, is to the Gain of
the one, and ^the Lofs of the other.

Therefore Multiplying i38x, difference of the numbers
expreffing the Odds, by 100, which is the fum ventured by
each Man, and dividing the product by 1384 fum of the num*
bers expreffing the Odds ; the Quotient will be 99 Guineas,

and about i8^'^~4t^ which confequently is to be
eftimated as the Gain of A.

REMARK IL

IF the number of Stakes which are to be won and 4ofl be
unequal, but the number of Chances to win and lofe be
equal ; the Probabilities of winning will be reciprocally pro-
portional to the number of Stakes to be won.

Thus, If A ventures Ten Stakes to win One ; the Odds
that he wins One before he lofes Ten will be as 10 to i.

But ten Chances to win One, and OneCliance to lofe ten,
makes the Play perfe£lly equal

H There-

iVOTWftKlfVnMnn^MMi .OTUt I ■ ■^^••••^^^li

26 The Doctrine 0/ CnANces.

Therefore he that ventures many Stakes to win but few,
has by it neither advantage nor diiadvantage.

PROBLEM X.

TfTO CameJlerSj A and B lajfyz^. Counters^ and fly with
Three Dice^ on this condition ; that if ii Points come ttfj
A [ban take one Counter out of the heap ; if i^ BjbaS take out
one J and be jb all be refuted to winj whofballfoonejl get ix Counters.
What are the Probabilities of their winning f

This Problem difTers from the preceding in this, that the
play will be at an end in xy Cafis of the Dice at mod, (that
is of thofe Cafts which are favourable either to ^ or JB : )
Whereas in the preceding cafe, the Counters paffing conti*
nually from one Hand to the other, it will often Happen that
A and B will be in fome of the fame circumftances diey
were in before, which will make the length of the play un-
limited.

SOLUTION.

TAking 4 and b in the proportion of the Chances that
there are to throw 11 and 14, let 44- ^ be raifed to
the x'^d Power, that is to fuch Power as is denoted by the
number of all die Counters wanting one : Then (hall the ix
firft Terms of that Power be to the ixlaft in the fame pro-
portion as are the refpedive. Probabilities of winning.

PROBLEM XL

•

THree Perfons AyB^ C out of d heap of ix Counters^
whereof four are White and Eight Blacky draw blindfold
one Counter at a time in this manner ; A begins to draw ; B foU
lows A i C follows B ; then A begins again ; and thej continue
to dram in the fame order^ till one of them^ who is to be reputed
to win, draws the firfi White one. What are the Probabilities of
their winning f

SOLU-

L

The Doctrine 0/ Chances. 27

SOLUTION.

E T II be the number of all the Counters, a the number
of White ones, b the number of Black ones, and x the
whole Stake or the fum playM for.

I* Since A has m Chances for a White Counter^ and t-
Chances for a Black one, it follows that the Probability of

his winning is ^j^ or -^J Therefore the Expeftation he ha»

upon the Stake i arifing from the circumftance he is in wheo
he begins to draw is ^ x i = •-- Let it therefore be a*

greed amongft the adventurers that A ihall have no Chance
for a White Counter, but that he ihall be reputed to have had
a Black one, which (ball actually be taken out of the heap,
and that he ibail have the fum -i. paid him out of the Stake

for an Equivalent* Now ^ being taken out of the Stake^

there will remain 1 — -^ = -i-=^ = — .

■ • •

X* Since B has 4 Chances for a White Counter, and that the
number of remaining Counters is n — i, his Probability of
winning will be .jl^ Whence his Expeclation upon the re-

maining Stake ^, arifing from the circumftance he is now in,.
will be ——. Suppofe it therefore agreed that B (hall have

the fum ^^ paid him out of the Stake, and that a Black
Counter be likewife taken out of the heap. This being done,
the remaining Stake will be — — ^ ^^J_ ^ % or "^Jj^^r/V but

nb — nb-zubbi Wherefore the remaining Stake is ***^*

■««— 1

3* Since C has 4 Chances for a White Counter, and that
the number of remaining Counters is n — x, his Probability
of winning will be -^^i And therefore his Expeftation up-

on the remaining Stake, arifing from the circumftance he is
now in, will be t"" iTnl!^. which we will likewife fup-
pofe to be paid him out of the Stake.

•4* ^ may have out of the remainder ,V/.r^x»i7x«'l-^ 5
and fo of the reft till the whole Stake be exhauftod.

There.

" "J ' v. P ^ I Mi l l » ■ I

^>iw>iiii*w^iipnpwwffiww<y<iyp^^^^^y^ I . jii j iTW^Bymf^iw^f^

fi8 Tt)e Doctrine 0/ Chancbs.

Therefore having written the following general Scries, viz.
-^+ ;r^,P + —a+ r^R + ^S &a wherein P,Q.,

R, S &c. denote the preceding Terms, take as many Terms
of this Series as there are Units in ^+1, (for (ince b repre-
firnts the number of Black Counters, the number of drawings
cannot exceed ^+ z ) then take for A the firft, fourth, ft-
venth &C. Terms; for B take the fecond, fifth, eighth &c.
Terms ; for C the third, (ixth &c and the fums of thofe
Terms will be the refpedive Expeftations of ^, 6, C ; or be-
caufe the Stake is fix^d, thefe fums will be proportional to
their refpeftive Probabilities of winning.

Now to apply this to the prefenc cafe, make ji=ix^
4 = 4,. ^ = 8, and the general Series will become

-ir + -n-P+-^a+TR + fS + -i^T + ^V + f X

+ -J- Y : Or multiplying the whole by 49^, to take away
ihe FraQions, the Series will be
j6y-|-ixo-4-84-|-56 + 3^ + ^^ + '^ + 4 + '•

Therefore atfign to A 165 + jtf-j- '0=*3^ i ^^ ^ ^*o
-^ 35 -^A = 159 } f o ^ 84 -1- 20 ^ I = lof, and their
Frobabilities of winning will be as 23 1, 159, 105, or a€

77» f 3> 35-

If there be never fo many Gamefters Aj B» C, D &c whe-
ther they take every one of them one Counter or more; or
whether the fame or a different number of Counters.; the
Probabilities of winning may be determined by the fame ge-
neral Series.

REMARK I.

T

H E preceding Series may in anv particular cafe be
ibortenM s for if 4 is = i, then tne Series will be

-r X 1 4; I + I +.1 + 1 + 1 + 1 &C.

Hence it may be obferved, that if the whole number of
Counters be exactly divifibl^by the. number of perlbns con-
cerned in the Play, and that there be but one (ingle White
Counter in the whole, there will be no advantage or difad-
vantage to any one of them from the iituation he is in, in
cefp^ to the order of drawing.

If

fg^rmim-^msmmmmm^wfffiffmmm^ f^ W *J — ji'i " W ».|. '"'''*''<^^!^^^W*

The Doctrine o/Chancesl a^

If if = 2. then the Series will be

ifX

If if := J. then the Series win be

If 4 = 4. then the Scries will be

„x,^,xJ^-»X»--3 XW— 1XK-.2XW— 3+«-2X»— 3XH— 48CC

Wherefore rcjc£ling the common M ultiplicators ; the (e-
vcral Terms of thcfe Series taken in due order will be Pro*
portional to the fevcral Expcftations of any number of Game-
flers. Thus in the cafe of this Problem where n\s-=z \x^
and M:=ibi the Terms of the Series will be

For A. For B. For C.

II X10X9=:990
8x7 X6=: Jj6

S X 4 X 3 = 60

10x9x8 = 71019x8x7 == 504

7 x6xy = xio
4 X3 XX = 14

6xfX4=^ x*^
ixxxi == 6

1386 954 630

Hence it follows, that the Probabilities of winning win be
rcfpcftively as 1 386, 9^4, 6;o ; or dividing all by 18^ as 77^
53, 3^9 as bad been before determined,

REMARK IL

BU T if the Terms of the Series are many, it will be con-
venienc to fum them up, by means of the following me-
thod, whofe Demonllracion may be had from the Methoius
DiffertntUlis of Sir Ifdsc Newtottj printed in his Analjjis.

Subtraft every Term, but the firft, from every following
Term, and let the remainders be called j&y? Differences \ fub«
trad in like manner every^firft difference from the following,
and let the remainders be called feconi, Differences ; fubtraft
again every one of thefe fecond differences from that which
follows, and call the remainders /i^/ri Differ enus\ and fa on,
till the la(l diiTerences become equaL Let.the firft Term be
called d^ the fecond h \ the firft of the firft diiTerences iT, the

I firft

'■ ■ " ■'* .uw.jii I •iM«^aMig«pvipq||iMt««^iiLLii. I ifi9w,««n|QpnnM*w«^nwmgM

30 The Doctrine o/Ci!ANCEs.

fiift of the fecond ditferences /', the firll of the third diflPe*
fences J!" &c. and let the number of Terms which follow the
/irft be x^ then will the fum of all thofe Terms be

-f X '-fi X ^^ X '^^'^ &c.

N. B. If the numbers whofe fums are to be taken are the
Produftsof two numbers, the fecond diflferences will be equal;
if they are the Produds of three, the third differences will
be equal) and fo on. Therefore the number of Terms, which
are to be taken after the firft, is to exceed only by Unity
the number of FaQors that enter the compofition of every
Term.

It may alio be obferved, that if thofe numbers are decrea*
Hng, it will be convenient to invert their order^ and make
that the firft which was the laft.

Thus, fuppofing the number of all the Counters to be loo,
and the number of White ones 4 : Then the number of all
the Terms belonging to ^,B, C will be 97, the laftof whkh
3 X X X I will belong to ^, fince 97 being divided by 3, the
remainder is i. Therefore beginning from the lowed Term
3 X X X i> and taking every third Term, as al(b the differen-
ces of thofe Terms, we Ihall have the following Scheone

3X ax I == 6

6x 5X 4 == ixo

384
9X 8x 7 = 504 431

816 i6x

11X11X10=: 1310 594

If Xi4Xi2 = x73<^

1410

From whence the Values of 4, *, d\ J% d'\ in the ge-
neral Theorem, will be found to be refpeflively 6,ixo, 384^
43 Xi t6xi and conlequently the fum of all thofe Terms wiD

DC

p^giffJ^'Py^MW^W 1 1 ■■< ; m i juj i l H l | p,|| |L P ^ W ]M ■ HJ^J «Aj i | i u i|, , ^ ,^ , 1 „ , ^ I . , ,^,mm, J ^^m m iw

The Doctrine o/Chances. 31

+ -f x^xiy2 xi^ xi6i, or
^ + J«T*+ 50-^* + 314-** +(J-|-x*, or

-^ >^ » + I X *+2 X '3X+I X S^+T

In like manner it will be found, that the fum of all the
Terms which belong to 5, the laft of which is 5x4x3
is

And alfo that the fum of all the Ternis belonging to C, the
laft of which is 4 X 3 x x, is

-^ X i+1 X x+2 X 9**+ 27*+ 16b

Now X in each cafe reprefents the number of Terms want*
ingone, which belong feverally to J^B^C; wherefore ma-
king X'\'i = p, their feveral Expectations will be refpe£tive-»
ly proportional to

f X p^i X 3£~x X 3^+1

f X ^+1 X 9^f+9^— 1-

Again, the number of all the Terms which belong to them
all being 97^ and A being to take firfti it follows, that/ in the
firft cafe is = 33, in the other two = 31*

Therefore the feveral ExpeQations of A^ By C will be re-
fpe£lively proportional to 4111^9 3959^i 38008.

If the number of ail the Counters were 500, and the num-
ber of the White ones fiill 4 ; then the number of all the
Terms reprefenting the Expefiations of ^^ B^C would be 497.
Now this number being divided by 3, the Quotient is 165^
and the Remainder x : From whence it follows, that the laft
Term 3 x i x 1 will belong to Bf the laft but one 4 X3 Xx
to Jy and the laft but two to C; it follows alfo, that for B^
and Jy f muft be interpreted by 166^ but for C by 165.

The

l im u !■ \m ■ \\t ,iw} \wFf^mim'mm9^rmmm'>mmm^^mmw^^^^f^m^Hlfli^

i;^

32 The Doctrine 0/ CiiAKcrs.

The GAME of BASSETE

RVLES of the PLAT.

TH E Dealer, otberwife called the Ba^ker^ 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 Poftttf has 1 3 Cards in his
hand, one of every fort, from the King to the Ace, which
I ) Cards are called a Book ; out of this Book he takes one
Card or more at pleafure, 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 firit cafe being particular (hall be calculated by it felf^
but the other two are comprehended under the fame Rules.

Suppofing the Ponte to lay down his Stake after theP^ck is
turnra, 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, be wins a Stake equal
to his own*

If the Card upon which the Ponte has laid a Stake comec
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 ukes his owa 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^ a half;
which to make the calculation more general we will call j.
In this cafe the Ponte is faid to be FdcesL

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 la(V of all, there is an exception frooi
the general Rule : for tho' it comes out in an odd place which
fliould intitle him to win a Stake equal to his own, yet he
neither wins nor lofes from that circumftance, but takes back
Jiis own Stake.

PRa

l7WgWfywq^lW**!"^< " .J H " II II ■i .J WH U W J . J tm ii ■■Lii .w. ^ <■<!■ . ' ■■ P . < |.tll 'I U '

7i&^ Doctrine o/ Changes. 35

PROBLEM XII.

TO EJlimate at Baflctc the lojs of the Fonte under am chr^
cumfimces of Cards remaining in the Stock, when he Itjs
his Stake^ and of any number of times that his Card is refeated
in it.

The Solution of this Problem containing Four Cafes, vizi
of the Fonts Card being once, twice, three or four times ia
the Stock ; we will give the Solution of all thefe Cafes feve-
rally.

S O L U T I O N of the firft Cafe.

THe Ponte has the following Chances to win or lofe, ac-
cording to the place his Card is in.

Chance for winning o

Chance for lofinp j

Chance for winnmg i

Chance for lofing i

Chance for winning i

Chance for lofing r

Chance for winnit^ a

It appears by this Scheme that he has as many Chances
to win I as to. lofe z, and that there are two Chances
for neither winning nor lodng, viz. the firft and laft, and
therefore that his only Lofs is upon account of his being
Faced: From which ^is plain that the number of Cards co>
vered by that which is in view being called n^ his Lois wiU

SOLUTION of the fccond Cafe;

By the firft Remark belonging to the Xlth Problem it appears
that the Chances which the Fonte has to win or loie arc
proportional to the numbers, n — i^n — i, n — 3 &c. There-
fore his Chances for winning and lofing may be exprclfed
by the following Scheme.

41 >

6 I
* 1 I

■|i«*apM«m«f*«PMWMM

n iiwf M iiii ii i ii" I I I ' * ^"" ' jt iM ft ".J.j ^y jj. wp

34

The Doctrine cf Chances.

I

»•

X

«■

3

n

4

a

5

m

6

n

7

\»

8

1

n

9

n

*

-I Chances
-X Chances

3 Chances
-4 Chances
-y Ctianccs
-6 Chances
-7 Chances
-8 Chances
-9 Chances

1 Chance

for winning
for lodng
for winning
for lofing
for winning
for lofing
for winning
for lofing
for winning
for lofing

Now ferting afide the firft and fccond number of Chan*
ces, ic will be found that the difference between the ji
and ^h is= i, and that the difference between the ^th and
6th is = I. The difference between the jth and 9th alfo
is = I, and fo on. But the number of differences is !L=l3, and

the fum of all the Chances is ^ x ^* Wherefore the Gain
of the Ponte is "*j ; but his Lofs upon account of the Fdce

is w- 2 X jf divided by i x -^, or ~rHr? • Hence it may be
concluded that his Lofs upon the whole is
.TTivj-— or _i^ fupKing^ =-!•.

That the number of Differences is ^-zl will be made evi-
dent from two conliderations.

Fir ft, the Scries n — 3, /i— 4, » — j &c. decreafesio Arlth*
metic Progreflion, tlie difference of its Terms being Unity,
and the laft Term alfo Unity, therefore the number of its
Terms is ecjual to the firft Term »~) : But the number of
Differences is one half of the number of Terms, therefore the

number of Differences will be ^-UL

2

Secondly,. It appears by the Xlth Problem^ that the num-
ber of all the Terms including the two firft is always ^-4*1 ;
But t in this cafe is = x. Therefore the number of all the
Terms is ^— i, from which excluding the two firft, the num-
ber of remaining Terms will be >9r*3, and codcquently the
number of Differences will be s^.

That the fum of all the Terms is -^ x '-=-^ is evident alfo

from two different confiderations. Firft,

pmmmm mn t ^ »MW\M f . n

^w>iiHg^ww#?^^r^iS W P u. '» uM< ii j ii . w j.. ' 1 1 .^ T I II ■ I . ■ . ■ ■> ■ I ■

n^e Doctrine 0/ Chances. 35

Flrft, In any Arithmetic Progrcffion whereof the firft Term
is /I— I, tlie dittcrcnce Unity, and the laft Term alfo Unity^
the fum of the Progreflion will be JJ-x^

Secondly, the Series ^t^zix n — i-j-«— a+ii— 3 &c.

belonging to the preceding Problem, cxpreffcs the fum of
the Probabilities of winning, which belong to the fevera>
Gamefters in the ca(e of two White Counters, when the num-
ber of all the Counters is m It therefore cxprefles likewife
the fum of the Probabilities of winning, which belong to tho
Ponte or Banker in the prefent cafe : But this fum mud al-
ways be equal to Unity, it being a certainty that the Ponte
or Banker muft win ; liippofing therefore that m — i, n — x,
«— 3 &c is = S. we fhall have the Equation ^IJ^^ =: i »

Therefore 5= -rx^'.

SOLUTION of the third Cafe

By the firft Remark of the XI//Sr Problem it appears that
the Chances which the Ponte has to win and lole, may be.
exprefled by the following Scheme^

I

X

3

4

5
6

7
8

n — I X n — X for winning o

n— X. X n — 3 for lofin^ 7

n — 3 X n — 4 for winning i

n — 4 X »— y for lodn^ 1

n — S X n — 6 for winning 1

n — 6 X n — 7 for lofmg r

^—•7 X »— 8 for winning k

8 X — 9 for lofing i

XX I for winning i

Setting afide the firft, fecond and laft number of Chances; .
it will be found that the difference between the rd and 4\th
iS zn — 8 ; the difference between the $th and 6tb xitr^- i %^
the difference between the ^th and Ztb xn — 16. &c Now
thefe differences conftitute an Arithmetic Progreflion, wherc«.
of the firft Term is x» — 8, the common difference 4, and
the laft Term 6^ being the difference between 4x3 and 3 x i»
Wherefore the fum of this Progreflion is i=i x i^, to wbicli

adding the laft Term xXi, which is favourable to the

Ponte,

III I »■ ■ I I M n mmiifiiwi iL i i ■ I I I I I II "T — ^nm

3^ The Doctrine 0/ Chances.

Fonte, the fum total will be ^ k^ But the fum of all the
Chances is ^ x ^ x ^ ; as may be concluded from the firft
Remark of the precedmg Problem: Therefore the Gaio of
the Pome is J^^^J!!!!!. * But his Lofs upon account of

the Face is j^^tI^tt^- Confequcmly his Lofs upon the
whole will be ^ii»E^i5L»r7'c/rJ«=^i«E} ^^ liLug —

wiiuAw will krw ax«x» — ix»— a axsxjf— iXM-2'

Suppofuig y — ^

SOLUTION of the fourth Cafe.

The Chances of the Ponte may be exprefled by the fol-
lowing Scheme.

I
%

3

4

5
6

7

n — 1 x^ — X x«— J for winning o

n-^x XH — J x« — 4 for lofing jr

H — 3 xn — 4 x« — 5 for winning i

^ — 4 x/>— s x»— ^ for loGng x

n — 5x/>— 6xJ» — 7 for winning i

n — 6x/i— 7X«— 8 for lofinp^ i

xn — 8xi» — 9 for winning i

3X ix 1 for lofing i

Setting afide the firft and fecond numbers of Chances, and
taking the ditfercnccs between the '^JL and /^th^ ph and 6tb ;
7th and Zth^ the lafl: of thefe differences will be found to be 1 8.
Now if the number of thefe differences be ^, and we begin
from the laft 189 their fum, from the fecond Remark of the
preceding Problem, will be colle£led to be ^x^ 4-i >^4 ^-4-5 :
And the number f in this cafe being ^=.% the lum of thefe

differences will be ^x^x^. But the fum of all the

wherefore the Gain of the

Chances is JLx^^x^^x — »

Ponte is

»x»— I x»— ax»— 3 >

now bis Lo(s upon account
of the Face is "7 "" VJ "" Tx^'LV and therefore his Lofs

m >c »— 1 X ■— a x.»— 3

upon the whole is
making jr =-i^

u X »— i X »— a

or

3 »— o

There

^HPigfWffW*^y^BWPWW**«i^^**l III ■ Ij i ii p ff!

^T^*^wi'^^rir:i^w*'i» .i. i . 1 ■

" * i w u i i» ' 'wwHwr— ■

" f" ^" w»1' q

Th Doctrine (/Chances. 37

There flill remains the finglc Cafe to be conHdcrcd, viz. what
theLofsof ihcPontcis, when he lays a Stake before the Pack is
turned up ; but there will be no diiliculty in it after what we
have faid, the diiTcrcncc between this Cafe and the reft being
only that he may be Faced by the firft Card difcoveredi which
will make his Lofs to be vl=A that is, about -rrJ:? part

of his Stake.

Thofe who are dePirous to try, by a kind of Mechanical
Operation, the truth of the Rules which have been given for
determining the Lofs of the Ponte in any Cafe, may do it in the
following manner. Suppofe for Inilance it were required to
find the Lofs of the Pome when his Card is twice in the Stock,
and there are five Cards remaining in the hands of the Ban-
ker befide the Card in View. Let them be difpofed accor-
ding to tills Scheme.

». 2, 3, 4, 5

* ♦ • • •

* • • * •

♦ • • • ♦

• • • ♦

• • ♦ * •

Where the places filled with Afterifcs [hew all the Vaiioos
Pofitions which the Ponte^s Card may obtain ; It is evident thac
the Ponte has four Cliances for neither winning nor loHng, three
Chances for the Face or for lofing -L, two Chances for winning

I, and one Chance forlodng i ; and confcquently that his Lois
is -^ to be diftributed into 10 parts, the Dumber of all the

Chances being io> which will make his Lofs to be -JL.. Like-
wife if the number of Cards that are covered by the firft were
fcvcn, it would be found that the Ponte would have fix Chan*
ces for neither winning nor lofmg, five Chances for the Faoe,
four Chances for winning i, three Chances for lofms i, two
Chances again for winning i, and one Chance for lofing i,
which would make his Lofs to be -ij-. And the like may be

done for any other Cafe whatfoever.

L From

' ' ' " ■ • ■ ' ■ I . ^wi I iiwmmi^mfm^^'v^mtm^^wsi^immmi^^mmm'mm^fi

fc*J^rifehll*U^ .A. .

38 The Doctrine 0/ Chances.

From what has been faid, a Table may eafily be compo-
fed, (heu'ing the feveral Loflcs of the Poiuc in whatever cir«
cumflance he may happen to be.

A TABLE for BASSETE.

N
52

I

^ >|C ^

2

3

4

* ^ *

^ ^ ^

•735

866

49
47

* * *

* ^ ^

867

98

23S2

1602
»474

801
737

94

90
86
82"

ir

74

2162

45

43
4»

1980

1351

1806

1234
1122

101$

617

1^40

561

__507__

457'

409

T63"

1482

1 190
ios;6

$lo

812'

702
600

?7

9<4

35

70

818

33

66

31

62

^42
5<52

487 ,,
418

354

^ 321

"281

243

209

29

58

27_
25

54

SO

23

46

S06

'77

»47'

31

42
38

420

~ 34'2
272

295

19

243
"" »94~

121

— -iilZ
75 ^

57 ""

>7

34

15

30

210

IJI

13
II

9

7

26

"4

22

no

72

82

41

28

17

18

5«

U

42

3?

The Ufe of this Table will be belt explained by one or
two Examples.. Exam-

^gypig!l^w^"rwqwiiWiwiWfByw!Wf*i<g^ iM ' m^ i i i 1 1 w

■ ^ii'ii u ii»i i iaifw^if^ww

T'^e Doctrine o/Chances. ^

EXAMPLE I.

LET It be propofed to find the Lofs of the Pontc when
there are x6 Cards remaining in theStock| and his Card
is twice in it.

In tlie Column N find the number ly, which is Icfs by
one than the number of Cards remaining in the Stock: Over
againft it, and under the number x, which is at the head of
the fccond Column, you will find 600; which is the Denomi-
nator of a Fraftion whofe Numerator is Unity, and which
iliews that his Lofs in that circumftance is one part in fix
hundred of his Stake.

EXAMPLE IL

TO find the Lofs of the Fonte when there is eight Cards re-
maining in the Stock, and his Card is three times in it.

In the Column N find the number 7, Icfs by one than
the number of Cards remaining in the Stock : Over againft 7,
and under the number 3 in the third Column, you will find
3 s i which denotes that his Lofs is one pare m thirty five
of his Stake.

CoroBdry I. 'Tis plain from the conftruftion of the Table,
that the fewer Cards are in the Stock, the greater is the LoC»
of the Ponte^

Corcllafj IL The leaft Lofs of the Ponte, under the fame
circumftances of Cards remaining in the Stock, is when hir
Card is but twice in it ; the next greater when three times ;
flill greater when four times, but bis greateft Lofs when ^is
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 ac-
cordingly, but it would be eafie to compofe other Tables to an-
fwcr that Variation, fince the quantity/, which has been af>
fumed to reprefent that Lofs may be interpreted at pleafure.
For in{lance,when the Lofs upon the Face is ^it has been found

in the Cafe of 7 Cards covered remaining in the Stock, and the
Card of the Ponte being twice in it, that his Lofs would be -^,

but upon fuppofition of its being ^ it will be found to be -^.

The

W >^ I

f .w i l u i iMwi ip ^n I . ■ ■ n ■ ll ^ J ^ n | ^J | | |^y^^w^p^^pr^ njm x^ u mt m i^t^wmm^gmm^sfm^

40 The Doctrine of Chances.

The GAME of PHARAON-

THE Calculation for PharAon is much like the prece-
ding, the renfonines about it being the fame; there-
fore I think it will be (ufficicnt to lay down the Rules of
the Play, and the Scheme of the Calculation.

RVLES of the PLAT.

Firfi^ The Banker holds a Pack of 51 Cards.

Seco^dljt He draws the Cards one after the other, laying
them alternately to his right and left hand.

Vnrilj^ the fonte 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, which are commonly called Puh.

Fourthly J The Banker wins the Stake of the Ponte, whea
the Card of the Ponte comes out in an odd place on bis
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 in
the iame Pull the Card of the Ponte comes out twice.

Sixthljy When the Card of the Ponte, being but once ia
the Stock, happens to be the laft, the Ponte neither wins
nor lofes.

Seventhly^ The Card of tlie Ponte being but twice in the
Stock, and the two lad Cards happening to be his Cards^
he then lofes bis whole Stake.

PROBLEM XIIL

To Find dt Pharaon the Gaih of the B^nker^ h my Cir^
cumflance of Cards remAinlng in the Stocky diid of the
number of times that the Ponte'^s Card is contained in it^

This Problem, containing four Cafes, that is, when the
Card of the Ponte is once, twice, three or four times in the
Sto:k,- we flial! give the Solution of thefe four Cafes fevc-
rally.

SOLU.

m JUmm^v im^'T * ^ ^^ '■ " »■ " ■ " "' ■■ iw O '

tmm^^^^m

The Doctrine 0/ Chances.

SOLUTION of the firft Cafe:

4«:

The Banker has the following number of Chances for win-
ning and lofingy viz.

I

X

3
4
S

I Chance for winning
I Chance for loHn^
I Chance for winning
I Chance for lofing
X Chance for winning

^ [ I Chance for lodng

I
I
t

■ X
X

o

Therefore the Gain of the Banker is JL. Suppofing m to
be the number of Cards in the Stock.

SOLUTION of the fecond Cafe ^

The Banker has the following Chances for winning
lofingi vtJu

and

I

%

5

4

6

7
8

{n — X Chances for winning i
X Chance for winning /

Chances for loHng

I

{/I— 4 Chances for winning x

I Chance for winning y

/r— -4 Chances for lofing x

{n — 6 Chances for winning x

I Chance for winning jr
Chances for lofing

{n — 8 Chances for winning i
X Chance for winning j
jir— -8 Chances for lofing t

\nmmt

:ic 1 I Chance for winning i

Therefore theGainofihe Banker is "^icV^'*'

fuppofing y=.-^

M

^imm^

— 1 f^* «XS«<i^

The

'-■*»'*«**w«^w^^fppnpp^9^r*ijiPF«"'ini5WWT^^^

42 Ihe Doctrine 0/ Chances.

The only thing that deferves to be explained here. Is this ;
how it comes to pafs that whereas at BAJfete the firft num*
bcr of Chances for winning was reprefented by n — i, here
'cis reprefented by n—x. To anfwer this it muft be remcm-
bcr'd, that according to the Law of this Play, if the Pontc's
Card comes 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 — 1 is divided
into two parts n — x and i, whereof the (irft 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 his winning the half of it.

SOLUTION of the third Cafe

The number of Chances which the Banker has for win*
Ding, and lofing are as follow;

1

3
4

5
6

{

X x;— } Chances for winning i
1 X n^^% Chances for winning, y
n — X X n — J Chances for lodng i

{n — 4 X n — 5 Chances for winning i

X X ^*-4 Chances for winning j

n — 4 X n — 5 Chances for loHng i

{n — 6 X n — 7 Chances for winning i

X X n^-6 Chances for winning j

n — 6 X n — 7 Chances for loHng i

{n — 8 X n — 9 Chances for winning
X X n — 8 Chances for winning
XX I Chances for lofing

I
J

Therefore the Gain of the Banker is ^Z— ^ or

J^

4xa — 1

fuppofmg j — \.

The number of Cliances for the Banker to win is divided
into two parts, whereof the firfl exprefles the Chances he
has for winning the whole Stake of the Ponies and the fe-
cond for winning the half thereof.

Now

jHBWyr^wiii ^i i., P n^j| i |. !

IV

JJVM'i "'■«*"*'

l^««^i|np«|l*OT«*'^N«>Wff*M>WNP«PWPP««M

■ *iiM w m-m

The Doctrine qf Chances.

43

Now for determining exaflly thefe two parts, it may be
confidcrcd, that in the firft Pull the number of Chances for
the firft Card to be the Ponte's is n — i x n—z ; alfo that
the number of Chances for the fecond to be the Ponte's but
not the firft, is n — x x n — 3 : Wherefore the number of Chan*
ces for the firft to be the Ponte's and not the fecond, is like-
wife n — z x n — 3. Hence it follows, that if from the num-
ber of Chances for the firft Card to be the Ponte^ viz. from
ft — I X n — X there be fubtrafted the number of Chances for
the fii ft to be the Ponte's and not the fecond, x^/^. n — x x »— 3,
there will remain the number of Chances for both firft and
fecond Cards to be the Pontc's, viz. x x «— x and fo for the
reft.

SOLUTION of the fourth Cafe.

The number of Chances which the Banker has for win*
nlng and lofing, are as follows \

{n — X X n — 3 X n —4 Chances for winning i
3 X »— i' X n — 3 Chances for winning j
n — X X «— 3 X n — 4 Chances for lofing x

3

4

S
6

7
8

{n — 4 X ff^^s ^ ^ — ^ Chances for winning 1
3 X n — 4 X 9t — 5* Chances for winning j
n — 4 X n — s X n — 6 Chances for lofing i

{

H — 6x/i-

3X»-

n — 6xn

7Xn^
6x«-
•7X»'

- 8 Chances for winning i
7 Chances for winning j
•8 Chances for lofing x

{

»— 8 x» — 9 X» — 10 Chances for winning x
•8x» — 9 Chances for winning jf
9 X /^ — I o Chances for lofing x

3X»
n—Z x«

{

XXIX

3 X X X

X X X X

Chances for winning x

1 Chances for winning x
o Chances for lofing x

2«-5

Therefore the Gain of the Banker is ^ztt^^ or ^^^^^ ^^_
fuppofingjp=^

A TA-

\

'^yw'^wwi I II nmn i j i LUj i M"

■I LL l ..l.p ip ij U ll .l ll im i^ lH III PM. I I I l ll HM.Uj^^ l ji. 1 1

h^w^i^Mi^ai

4f

The Doctrine 0/ Chancei

A TABLE for PHARAOU,

N I

2

3

4

52

so

48
46

44
42

;jc ^ ^

'fC ^ ^

5(C ^ 5|C

50

* * *

94

*5

48

48
44

90

62

4f

44

42

85
82

74

5o

__57

_~54"_

42
"40'
'38

36

40 __
"38"
3*
34
32
30

40

52

49

38

__ 70 __
"55"

36

45
44

34

34

tf2
58

32

32

4»

38

36

"33^

3o'__

28"

25
22

20
17

30

30

54

28

28

28

50
"46"

25

24

26

26

24

24

4»

22

20

22
20

22

38

34
30

25

20

18
r5
14

18

18

16

16

H

H

22

12

10
8

12
10
8

12

18

12
9

10

»4

8

II

6

The numbers of the foregoing Table, as well as thole of t
Table for Bijfetej are fufficiently exa£k to give at firft view
Idea of the advantage of the Banker in all circumftances : £
if an abfolute degree of exa6lnefs be required, it will be eaG
obtained from the Rules given at the end of each Cafe.

PR

WH^iWBH— 1— wuCTP"! ■ i JW.Wif ii m i m ilii L i m ■^ ■ . ■ F". ' '^ .pp».i Pi M i HP i J ii n p p.iij i

9»* ■ in.U

The Doctrine b/ Chances. 45

PROBLEM XIV.

1/^ A, B, C throw in their turns a reguldr Bdll^ hdving four
White Faces and eight Black ones; and he be to be refuted to
win who /ball frjl bring uf one of the White Faces : It is demand*
ed what is the proportion of their refpeffive Probabilities of mn»
ning f

SOLUTION,

THe method of rcafoning in this Problem is cxa£bly the
fame with that which we made ufcof in the Solution of
the Xlth Problem : But whereas the different throws of the
Ball do not diminifli the number of its Faces; in the room of
the Quantities b — i, *— x, b — 3 8:0, n — 1» »-— x, n — 3 &c.
employed in the Solution of the aforefaid Problem, we muft
fubilitute b and n refpe£lively, and the Series belonging to
that Problem will be changed into the following, viz.

which is to be continued infinitely : Then taking every third
Term thereof, the refpe£tive Expectations of A^ B, C will
be expreffed by the three following Series.

mh , mh* . mil % si'* . tf^"

But the Terms of which each Series is compounded are m
Geometric Propreflion, and the Ratio of each Term to the
following the ^me in each of them; Wherefore the Sums of
thefe Series are in the fame proportion as their firft Terms,

yi;gz. as -^, -^, -^ or as nnj bn^ bb ; that is, in the prc-

fent Cafe, as 1-14, 96^ 64, or as 9, 6, 4. Hence the refpeftive
Probabilities ot m inning will be likewife as the. numbers 9,
6, 4.

Corollary I. If there be any other number of GameHers
A^ Bf Cy D &c playing on the fame conditions as above;;

N take

'^mrtm'^u I I 11 11 ■ ■^^^^^ 1 1 . i n ••'>yy^tyypMp. i j ■ 1 1 . w»iyjn^ygia^pi^^i^wMi^«qr'

4* The Doctrine of Chances.

take as many Terms in the Ratio of /i to ^ as there are
GameAers, and thofe Terms will refpeClively denote the fe-
veral Expectations of each Gamefter.

CorolUrj IL If there be any number of Gamcdcrs J^ B^
Cj D &c. playing on the fame conditions as above ; with this
difference only, that all the Faces of the Ball arc markM by
particular Figures, i,i, ), 4 &c. and that a certain number^
of thofe Faces (hall intitle W to be the winner ; and that like-
wife any other number of thsm, as jr, r, i, / &c. (hall refpc-
ftively intitle 5,C, D, E &c. to be wmners : Make n^p=id^
n — ^ = ti n — r-=.Cy i9-^s=zd, n — t =e &c then in the fol-
lowing Series,

"T" + "ITT + "S" + -"1?" + —^ ^^

the Terms taken in due order ihall reprefent the feveral
Probabilities of winning.

For if the Law of the Play be fuch, that every Man ha-
ving once playM in his turn, (hall begin again regularly ia
the fame manner, and that continually till fuch time as one
of them wins : Then take as many Terms of the Series as
there are Gamefters, and thofe Terms (hall reprefent the re-
fpedive Probabilities of winning.

And if it were the Law of the Flay, that every Man
fhould play feveral times together, for inftance twice : Thea
taking for A the two firft Terms, for B the two following,
and fo on ; each Couple of Terms (hall reprefent their refpe-
flive Probabilities of winning; obferving that now f and f
are equal, as alfo r and /.

But if the Law of the Flay (hould be Irregular, then you
muft uke for each Man as many Terms of the Series as will
anfwcr that Irregularity, and continue the Series till fuch time
as it gives a fuflicient Approximation.

Yet, if at any time the Law of the Play having been Ir-
regular (hould afterwards recover its Regularity, the Proba-
bilities of winning will (with the help of this Series) be de-
termined by finite expreflions.

Thus, if it (hould be the Law of the Play, that two Mea
A and B^ having play'd irregularly for ten times together,
(hould afterwards play alternately each in his turn: Diftri-
bute the tea firft Terms of the Series between them, accord-

log

The Doctrine o/Chances. 47

ing to their order of playing ; and having fubtraQed the fum
of thofe Terms from Unity, divide the remainder of it be*
tween them^ in the proportion of the two following Term%
which add refpe£lively to the (hares they had before : Then
fhall the two parts of Unity which ^ and £ have thus ob«>
tained, be proportional to their refpeQive Probabilities of
winning.

Of Fermutations and Combifiatiofts.

Permutations are the Changes which fcveral things can re-^
ccive in the different Orders m which they may be placed,
being confidered as taken two and two, three and three, four
and four, &c

Combinations are the various Conjun^ions which feveral
things may receive without any re(pe£l to Order, being ca«
ken two and two, three and three, four and four, &c

LEMMA. ,

TF the Probability that an Event {ball Haffen he -I-, smi if tia

Event being fuffofei to have Hafpenedj the PriAaHtitj of
another s Haffening be «2-; the Probability of both Haffemng mi

be ^x -^ or — ^. This having been alreadjf Demonfiraiei it^-

the IntroduSion^ mUnatrefuire any farther frocf.

PROBLEM XV.

AA^ T number of Things a, b, c, d, c, f being given, ant cf
which Two are taken as it hapfens : To find the ProhJbWSy
that any one of them^ as a, (ball be the firjl taken, and anj
other, as b, the fecond.

SOLUTION. ^

TH E number of Things in this Example being Six, it fol-
lows that the Probability of taking a in the firft place
is -i-: Let a be confidered as taken, then the Probability of

taking b will be JL; wherefore the Probability of taking firft

4 and then ^ b 4- )c -i-= -—«

' ' ^ CorcU

™"— ^~'"~~~-~~— 1 — n JT"T i i^ww'wii 1 .; i ■. ,j i . wii ^.ipjT r i ' ' "": — rrT"n~'— TTT-

48 The Doctrine of Chancss.

Corolisay. Since the taking of s in the firft place and t
in cl)e fccond, is but one tingle Cafe of thole by which
Six Things may change their Order, being taken two and
two ; it follows, that the number of Changes or Permutations
of Six Things taken two and two muft be jo*

GenerABjy Let m be any number of Things ; the Probabili-
ty of taking 4 in tlie hrft place and b in the fecond, will
be — ! — . and the number of Permutations of thofe Things

taken two by two will be 4v x

PROBLEM XVL

ANT number n cf TTfiftgs a, b, c, d, e, f being given ^ oat
of which Three dre taken ds it Hxffens : To find the Probs^
bilitj tbdt a fidll be the firfi tdken^ b the feeond and C the third.

SOLUTION.

'T'Hc Probability of taking a in the firft place is -i- : Let a be

^ confidcred as taken ; tbe Probability of taking b will be JL :

Suppofe both 4 and b taken, the Probability of taking c will
be -1-. Wherefore the Probability of taking firft 4, then b^

and\hirdly c, will be ^x-i-X^ =-ji^

CorollMrj. Since the taking of a in the firft place^ b in the
feeond, and c in the third, is but one fmgle Cafe of thofe
by which Six Things may change their Order, being taken
three and threes it follows, that the number of Changes or
Permutations of Six Things, taken three and three, muft be
6xf X4 = x%o.

CeneraBjy If ji be any number of Things; the Probabili-
ty of taking 4 in the firft place, b in the feeond and e in the
third win be — X --h- X -—• And the number of Permu-
utions of three Things will be n^n^t X

General COROLLART.

The number of Permutations of n Things, out of which
as many are taken together as there are Units m ^, will be
tf X »— I N i»— X X M — 3, &C. continued 10 fo many Terms as
there are Units in f.

Thus,

The Doctrine of Cnknttt. \9

Thus, the number of Permutations of Six Things takca
four and four, will be (J>^5^4X}= \6o. Likewifc the num.
bcr of Permutations of Six Things taken all together will be

' PROBLEM XVII.

TO Find the Probabilitj thst dny number cf Thingf, t»berecf
fome are repedted feverd ttmes^ fiaU dU be tdien in dwj
Order frofojed: For Infidnce^ thai aabbbcccc fjbdll be taken
in the Order wherein thej are written.

SOLUTION.

'npHe Probability of takings in thefirft place is -2-.: Sup-

pofing one a to be taken; the Probability of taking the
other is -|-. Let now the two firft Letters be fuppoled to

be taken, the Probability of taking b will be -1-: Let this

alfo be fuppofed taken, the Probability of taking another b
will be -i- : Let this likewife be fuppofed taken, the Proba*

bility of taking the third b will be -i.; after which there

remaining nothing but the Letter r, the Probability of taking
it becomes a certainty, and confcquently is equal to Unity.
Wherefore the Probability of taking all thofe Letters in the
Order given is -|. x-J-x-f-x -r-x^-.

Corollary. Therefore the number of Permutations which
the Letters aabbbcccc may receive, being taken all together
will be ^"^"7 . >^^M ,

Generalh* The number of Permutations which any num-
ber n of Things inay receive, being taken all together,
whereof the firit fort is repeated f times, the fecond q
times, the third r times, the fourth / times, &c will be thi
Series n x n — i x n — x x n — 3 x n — ^49 Src. continued to fo
many Terms as there are Units in ^ -|" ? + *" ^'' ^— J, divided
by thcProduftof the following Series, a//j&. ^x^ — i x^ — i,8rc.
X jT X q — 1 X J— X| &c. X r x r — i xr — x, S:c. whereof the
iiril mud: be continued to fo many Terms as there are Units
in f\ the fecond, to fo many Terms as three are Units in f $
the third, to fo many Terms as there are Units in r &c.

O PROB.

Ill i ii i^ Myw w i. 11 I ip.iii i . w . i — . ■■n^ff i iWi M iiwi W li' lW III I ji ^' " — 11. J. 'i . l . i l' ■■ ■ ■i ,H

50 T^ht Dpctrinb of Chancss.

PROBLEM XVIIL

AtJT number of Things a, b, c, d, e, f being given: TV
find the ProLxbility fhat^ in Uking trvo of them ds it msy
Usffn^ both a Mi b fbdl bt taken tndefendehtlj^ or withota
an; regard to Order.

SOLUTION.

TH E Probability of taking 4 or ^ In the firft place
will be -|-| fuppofe one of them taken, as for In*

(hnce d^ then the Probability of taking b will be r-^ • Where*

fore the Probability of taking both s and b will be ^ x -^

CorolLaj. The taking of both n and b is but one fingle
afe of all thofe by which Six Things may be combined
two and two ; wherefore the number of Combinations of
Six Things taken two and two will be ^ x -f- = > 5*

GemrdOj. The number of Combinations of n Things
taken two and two^ will be -4-x ~ '

PROBLEM XIX.

ANT number of Things a, b^ ^ d, c, f being £iven: Ta
find the Probabilitj^ that in taking three ^ them 4/ it
JHaffenSf tbej (ball be Anj three frofofed^ ^ a, b| c > no reffeli
being bad to Orditr*

SOLUTION.

TH E Probability of taking either 4, or h^ or c in the
firll place will be -|^ Suppofe one of them as 4 to

be taken, then the Probability of taking b^ or c in the feeond
place Mull be -|-. Again let either of them taken, as fup»

pofe b i then the Probability of uking c in the third place
will be -J^; wherefore the Probability of taking the three

Things propofedi viz. d^b^c will be ^« x ^ x -J-*

Corot-

P^yryTI W *■ " ? * " ^.wm ' ' ■ ■■ ■ m ww.^pwiwyt n ■■ » iT*^^Bff'<^f.wiwawrfww>»yfi»i|i ^u*i iy<wwi»p»i<« M ^'i * "

3

•* d -m

DOCTKINE of CHANCt>. ^

CofoUixy. The takiog of a^ b^ c is but one flnde Calb of
all chofe by which Six Things may be combined three and^
three ; wherefore the number of Combinations of Six Things
taken three and three will be -i-.x-^x-^=xa

Genet dl). The number of Combinations of /i Things Com*
bined according to the number /, will be

;xilli'x/laxi!^lx/lt' &^ Both Numerator and Denomi-
nator bemg concmued to fo many Terms as there are Units
in /»

PROBLEM XX:

To fni whit Protahi/ity there iSj that in taking ms it^'Hsfp*
Pens Seven Counters out of Twelve^ tpheretfjour srt Wbitt^
McL eight Blacky three cf them fig/l he White ones.

SO LUTION.

Flrfty Find the number of Chances for taking three White
ones out of four, which will be -i-x -^ x -^ = 4*
Secondfy^ Find the number of Chances for taidi^ four
Black ones out of eight : Thefe Chances will be found to be

I 1 I 4 '^^

Thirdly J Becaufe every one of the preceding Chances may
be joined with every one of the latter, it follows, that the
number of Chances for uking three White ones and four
Black ones, will be 4 x 70 = x8o.

Fourthly^ Find the number of Chances for taking four
White ones out of four, which will be found to be

Fifthly^ Find the number of Chances for taking three Bktck
ones out of eight, which will be -i-x-Z-x*-!-- = y5.

Sixthly^ Mukiply thefe two lad numbers together, and the
Produft j6 will Inew that there are y6 Chances for taking
four White ones and three Black ones ; .which is a Cafe noc
exprcffed in the Problem, yet is implyed : For he who under-
takes to take three White Counters out of eight, is reputed
to be a winner tho* he. takes fourj uolefs the contrary be cx^-
prefly ftipulated. Seventhly^

^'mr^i^'i^mmr^mmm

* m^ m ^} m\ w ^ i iiiii m i i m p^ m ). ■ i . i ■ t wj i>"WW'WWHyi|WB>Tf!

52 The DocTRiNB 0/ Chances. .

Sevefgfhfyj \Vhcreforc the number of Chances for taking
three White Counters will be i8o -|- 5^ ::= 3 }*•

Eighthly^ Seek the number of all the Chances for taking
fevcn Counters out of twelve, which will be found to be

ii K II K ioiC9)c8K7^fg ^ Q

1 % 6 ^ $ X ^x^x^ui 7y •

Laftly^ Divide the preceding number 336 by the laft 791,
and the Quotient -3^, or -^^ will be the Probability re-
quired.

CorolUrj. Let n be the number of all the Counters, 4 the
number of White ones, b the number of Black ones, e the
number of Counters to be taken out of the number /i; then
tlie number of Chances for taking none of the White ones,
or one fmgle White, or two White ones and no more, or
xhree White ones and no more;, or four White ones and -no
more, ice will be expreft as follows.

The number of Terms wherein b enters being always equal
to e-^a^ and the whole number of Terms equal to c.

But the number of all the Chances for taking a certain
number e of Counters out of the number n^ with one or
more White ones, or without any, will be

^X^XJ^^X^X^X^X^X^&C.

which Series mud be continued to fo many Terms as there
are Units in c.

REMARK.

TF the numbers n and c were large, facb as 40000 and
Sooo, the foregoing method would fcem impracticable,
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 aftually fct down, a great many of
them being common Divifors, might be expunged out of ooth
Series. However to avoid the trouble of fetting down fo
many Terms, it will be convenient to ufe the following The-
orem, which is a contraQion of that Method.

Let therefore n be the number of all the Counters, a the
number of White ones, cx\\^ number of Counters to be taken

out

■gyiBpl^PPfHI<jyf||Pjywpp|«>yyq|g"^ Hj mim iii ^twnnp—w^pwgir

The DoctkiNBo/CHANCfei 5j

out of the number n^ p the number of White Counters to
be taken precifely in the numbers: Then making ;y — ezsJU
I fay that the Probability of taking precifely the number f
of White Counters will be

cxc^l XC—X &C. X^X^— I Xd—X &C x-f xH^X^8{Ci

^Xff'-'ixn — xx«— JXi^'— 3Xir— 4Xi»— 5X*— 6 &C

Here it is to be obferved, ttiat the Nunierator confifts of
three Series, which are to be Multiplied together ; whereof
the firft contains as many Terms as there are Units in pj the
fecond as many as there are Units in d^p^ the third as many
as there are Units p : And the Denominator as many as there
are Units in s.

PROBLEM XXL

IW A Lottery confining of 40 coo Tickets^ dmanz iphich sre
Three particuUr Benefits: Wba is the ProtdtiTitf thdi tM»
kiffg 8000 cf them^ one cr mofe of the Three pmieutdf Beaefid
fbdll be dmongft them f

SOLUTION*

FIrft in the Theorem belonging to the Remark of the fore-
going Problem, having fubftituted refpe£lively, 80CO9
4000O1 31000^ 3 and I, in the room of r, ir, i^ d and p\
it will appear, that the Probability of taking (precifely one
of the Three particular Benefits will be
goooK uobox^i^yp^l or -i' iieark.

40000 X 39999 K 39998 '125 ^

S.coftdfyj c, n^ d^ A being interpreted as before, let us fup-
pofe p s=s %. Hence the Probability of taking precifely Two
of the particular Benefits will be found to be

Joooii^7mi3.\^U, or ^ very neat.

40000 X 39999 % 3999s 115 ^

Thirdly^ Makmg f =: 3« The Probability of taking all the
Three particular Benefits will be found to be

goooX 7999X7998 ^ _l_

40000x39998x39998 1 las ^ "^HM*

P Where-

a. fc I ■ ■

54 ^^^ Doctrine 0/ Chancer

Wherefore the Probability of taking one or more of the
Three particular Benefits will be ^^\[\^^ > or -^ very

near.

If. B. Thefe three Operations might be contraAed into
one, by inquiring what the Probability is, that none of the
particular Benefits may be taken; for then it will be found to

be ^'^^^'^ll^'US f = -rh nearly; which being fub.

64 ^^61
iiiucr I— -

the Probability required.

trafted from 1, the Remainder i p^, or -7^ will (hew

PROBLEM XXII.

TO ¥ini hotp mAnj Tickets ought be tiken in 4 Loitefy eoM^
fifliftg ^40000, dmong which there are Three fMrticu*
Imt Benefits^ to make it as ProhMe that one or tnore of tbojg
Three mg he takem as ttot.

SOLUTION.

LET the number of Tickets requifite to be taken be jc:
It will follow therefore from the Theorem belonging
to the Remark of the XXth. Problem, that the Probability
of not taking amongft them any of the particular Benefits
will be i^x ";!;■ X -^£7^* But tlmProbability is = -|-y
fince the Probability of the contrary is ^ by Hypothefis •
whence it follows that -i^x ""*"' X """"* ■ ss -i- This

Equation being folved, the value of x will be found to be
nearly 8x5 x*.

If. B. The Faflors, whereof both Numerator and Deno-
minator are compofed being in Arithmetic Progrefllion, and
the diflference being very fmall in refpefl of # ; thofe Terms
may be confidered as beins in Geometric Frogrefliony where-
fore the Cube of the middle Term -^r£=L may be fuppofed

equal to the Produfl of thofe Terms; from whence will a«
rife the Equation ^?' =• -i- or I^" =s -^ (ne-
gleAing Unity in both Numerator and Denominator ) and

GOO*

fff^yffywp n i I ■< iii L i I nui iw. j i ■ I 11 ■■ . 1 I I i i iii mi i . ii ■ ■! i. i i, I

The Dgctrinec/'Chancbs, 55^

ccnfequently x will be found to be nxi-*^;., but m-i^

= 4COOO, and i— v|. = 0.1063 ; Therefore x ss 82 ji.

In the Remark belonging to the V/A 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 lliould be taken, as not ; but in that Rule it b fup*
pofed that the proportion of the Blanks to the Prizes was
often repeated, as it ufually is in Lotteries: Now io the
Cafe of tlie prefent Problem, the particular Benefits being
but Three in all the remaining Tickets are to be confidered
as Blanks in refpeft of them; from whence it follows, that
the proportion of the number of Blanks to one Prize being
very near as i]))i to i-, and that proportion being repeat^
ed but three times in the whole number of Tfckets, the RuI6
there gi' en woulJ not have been fuSicienthr exa6l io this
Cafe ; to fupply which it was thought neceuary to give d»
Solution of this Problem.

PROBLEM XXHL

To Ftni df Pharaon, bow much it is tha the Bdnker geiM
per Cent of mU the Monej that is aivefttured^

HTPOTHESIS.

ISuppofe, /vry?, that there is but one fingle Pbnte : Secom^
Ijy That he lays his Money upon one Gogle Card at a
time : Thirdly^ That he begins to take a Card in the begin*
ning of the Game : Fourthly^ That he continues to take a
new Card after the laying down of every Pull: Fifthly^ TTiar
when there remains but bix Cards in the Stocky he ceafes to
take a Card..

SOLUTION.

WHEN at any time the Pome lays a new Stake up<^
on a Card tdken as it Happens out of his Book,
lee the number of Cards that are already laid down by the
Banker be fuppofed equal to x.

Now

iiMi * . »m ' m a l■ J ^^ p ■ l;p^ ^■■ ^ l | p^^^K^^p^^|piyy^^^li^^*»y^^^ . ' ■ W' . ^*'

.«ifcM«— ><1— I II ■ Mill I .1 I ~ ■» ■

5^ Tl&eDoCTHINBO/CHAMCtS.

Now in this clrcumftance, the Card taken by the Ponte
has cither paft four timcsi or three timcsi or twice, or once,
or not at alL

Firjl^ If it has palTcd four times, he can be no lofcr upon
that account.

Secondly^ If it has pafTcd three times, then his Card is once
in the Stocks now the number of Cards remaining in the
Stock being n—x^ it follows by the firft Cafe of the XWlth
Problem that the lofs of the Ponte will be -j^ : But by the

Remark belonging to the XX/i& Problem, the Probability
that his Card has palTed three times precifcly in x Card

is JL5i^i£Si!!5^. Now fuppofing the Denominator e-

qnal to /, Multiply the lofs he will fufTer ( if he has that
Chance ) bjr the Probability of having it, and the Produd

««x>ixx>2«4 will be his abfolute lofs in that circum^

^ •

fiance.

Tbirdfyf If it has paffed twice, his lofs by the fecond Caft

of the XllUb Problem will be Ji^iiiU but the Pro-
hability that his Card has palfcd twice in x Cards, is by the
Remark of the XXib Problem, .'^"'^^^"T'.T',^,^. ^

wherefore Multiplying the lofs he will fuifer (if he has that
Chance) by the Probability of his having it, the Produft

x^l^i^^m-^x-^f^6 ^iii be his abfolute lofs in that

drcumftance.

Fourthfy^ If it has paflcd once, his lo{s Multiptyed into
the Probability that it has pafled^ will make his abfolute loft

to be JL21»El2iEiEiJLL.

Fifthly^ If it has not yet pafled, his lofs Multiptyed idtd
the Probability that it has not paffed, will make his abfolute

lofs to be JEZ^ ''-»•'/ ^ ^ ' ^i=L.

Now the Sum of all thefe lofles of the Ponte^s will be

,»-»„H-y»-?>-f«>+3*^ .^ and this is the lofs ht fuffers by

venturing a new Stake after any number of Cards x are
pa{(.

JBut

• ,

^^•>»W!»^»«T'^:^T"w?wi^p»irip*i**wppw^aT!W!f^^

The Doctrine 0/ Chances. 57

But the number of Pulls which at any time are laid dowiip
is always one half of the number of Cards that arc paft;
wherefore calling t the number of thofe Pulls, the Lofs of the

Ponte may be exprefled thus, »^-T»»»'f y«^<^^-^^^-f m^'^

Let now f be the number of Stakes which the Ponte adven*
tures; let alfo the Lofs of the Ponte be divided into two parts.

And fince he adventures a Stake f times ; it follows, that the
firft part of his Lofs will be M^-ff;»-ff ^g,^

In order to find the fecond parr, let / be interpreted fuc-
ceflively by o, i, x, J &c. to the laft Term />— i ; Then in
the room of 6/ we (hall have a fum of numbers in Arhh-
metic Progreflion to be Muitiplyed by 6 ; in the room of
6 / / we (hall have a fum of Squares whofe Roots are in Arith*
metic Progrc(rion to be Muitiplyed by 6; and in the room
of 14/' we (hall have a fum of Cubes whofe Roots are in
Arithmetic Progre(fion to be Muitiplyed by 24 : Thefe (eve-
ral fums being collefted, according to the IW Remark on

the XI/A Problem, will be found to be ^p'-^^r-^^^rtA-^r ^
and therefore the whole Lofs of the Ponte will be

i

Now this being the Lofs which the Ponte fuftains by adven-
turing the fum />, each Stake being fuppofed equal to Unity,
it follows, that the Lofs fer Cent ofthe Ponte, or the Gain^

Cen t of the Banker is *' ^ i'^-Vv^^'^ft+a f •¥ » ^ ^^

or ^"-y + ^7'^^/^-*^^^ X ICO- Let

now n be interpreted by 52, and ^ by xs ; and the Gain of
the Banker will be found to be 2.99x^1, tnat is x /. 19^^- 10^
fer Cent.

By the fame MethoJ of arguing, it will be found that the
Gain fer Cent of the Banker, at Biffete^ will be

;«»-9 ^ 4f>^7^-^i J^L, X 100. Let if be

interpreted by 5 1, and f by 23 ; and the foregoing expreflion

Q^ will

■ iM i Bui .ii J i l l w , , iii,M ,„ i ft ^ m i l y i m ^ t yn . wt^mtmmu mmw ili JU fWWqwpwP

wm

$8 7£f DocTRiN£ of Chances.

will become o^po^Si, or 15^- 9 i. half-penny. The confi.
deration of the tirll Stake, which is adventured before the
Pack is turned, being here omitted as being out of the ge-
neral Rule: But if that Cafe be taken in, and the Ponte ad-
ventures 100/. in %\ Stakes, the Gain o\ th;? Banker will be
diminifhed, and becomes only 0.76145, that is, 15 ***• 3 i ve-
TV near: And this is to be eflimated, as the gain fer Cent of
the Banker when he takes but half Face.

Now whether the Ponte takes one Card at a time or fe-
vcral Cards, the Gain per Cent of the Banker continues the
fame : Whether the Ponte keeps conftantly to the fame Stake,
or fome times doubles or triples it, the Gain per Cent is flill
the fame : Whether there be but one linde Ponte or feve-
rai, his Gain fer Cent is not thereby altered. Wherefore the
Gain ftr Cent of the Banker of all the Money that is adven-
tured at Phardon ]% z L 19^^ 10 d. and at B^Jfete 15^ ^d.

PROBLEM XXIV.

SVpfoftng A dnd B to fldji together ^ the Chances thejf have re^
JfeHiveh to mn being as z toh^ dnd B obliging himfelf to
Set to A, Jo long as A wins without interruption i IVbdi is the
Advdntdge thdt A gets bj his Hand f

SOLUTION.
Irfi^ U A and B each Stake One^ the Gain of A on

the firft Game is -7x5- •

#-*

F

Secondly^ His Gain on the fecond Game will alfo be ^. ^^
provided he ihould happen to win the iirft : But the Proba-
bility of A^s winning the firft Game is j4j. Wherefore

his Gain on the fecond Game will be --Trr x -^^7 •

ThirMj^ His Gain on the third Game, after winning the two
firfty will be likewife -^x;* But the Probability of A^s winning

the two firft Games is '^T^ 5 Wherefore his Gain on the

third

■■ - ■■■ ■M I UW I PH I L,. . ■ ■■in . . .t i w ■ , .. ,M^ . j , „ , i i,.. ,^.^ w , . , . , lj„ , i . .. i,,,

The Doctrine 0/ Chances. 5^

third Game, when it is eftiinated before the Play begios.

IS -^^^Attt- X -^ &c

Founhfyf Wherefore the Gain of the Hand of ^ is an in-
finite Series, i;/x. I +-4t + 1^« + T^ + ^T^* ^^•
to be Multiplyed by •^^. But the fum of that infinite
Scries is -^^ j Wherefore the Gain of the Hand of A

IS -J— ^TFT - * •

CorolUrj I. If ^ has the advantage of the Odds, and B-
Sets his Hand out, the Gain of A is the difierence of the
numbers exprefiing the Odds divided by the lelTer. Thus-
if A has the Odds of Five to Three, then his Gain will

be -^^^= -^

CoroUarji II. If B has the Difadvanuge of the Odds, and*
A Sets his Hand out, the Lofs of B will be the difference
of the number exprefling the Odds divided by the greater ^
Thus if B has but Three to Five of the Game, his Lofs will

be -7-*

CoroUayy IIL If A and B do mutually engag^e to Set to-
one-another as long as either of them wins without inters

ruption, the Gain of A will be found to be ^^-^^^ • That

is the fum of the numbers exprefling the Odds Multiplyed
by their difference, the produQ of that Multiplication bemg
divided by the Frodu£l of the numbers exprefling the Odds.
Thus if the Odds were as Five to Three, the fum of $ and
3. is 8, and the difference x ; Multiply 8 by a, and the
Frodua x6 being divided by 15 (Froduft of the number

cxprefTing the Odds) the Quotient will be -Ji«^ or i-iTt
which therefore will be the Gain of A.

PROBLEM XXV.

ANT given number of Letters a, b, c, d, e, f &c. oB if item
different y being taken fromifcuoujfy^ ds it Ha f fens: To find
the t rob Ability that fome of tbim JbAS- be found in their piAces^,

Mtording^

mi l » ■■. ■ 1 1 ■■>■■■ ■ I , ii^i j p ^ ii i jn|^L iiii|iip .|| ii^ff>ip«wiitBpwWWWWWIIP^WWiMlWWH

6o T/je Doctrine 0/ Changes.

according to the rank they obtdin in the Alphabet ; and that others
of them /ball at the fame time be found out of their f laces.

SOLUTION.

E T the number of all the Letters be =s /r ; let the num
ber of thofc that are to be in their places be = p^ and
the number of thofe that are to be out of their places = f.

Suppofe for Brevity fake -i^ == r,

L

= /.

• xa-i *9 axa — iia— 2

= / 1 = V &c. then let all the Quantities

f , r, Sy /, V &C. be written down with Signs alternately po-
fitive and negative, beginning at i, if^ be =s o; at r, if
f zs I ; at /, \[ p=z % &c. Prefix to the(e Quantities the re-
fpeQive Coefficients of a Binomial Power, whofe Index is ss j:
This being done, thofe Quantities taken all together will ex-
prefs the Probability required ; thus the Probability that in
Six Letters taken promi(cuoufly, two of them, vijc. a and b
fhall be in their places, and three of them, viz. r, d^ e out of
their places, will be

6x5 6x5x4 ^^ 6x5x4x3 6x5x4x3x2 "" 720 '

And the Probability tliat a (hall be in its place, and ^, r,
Jf e out of their places, will be

J^ «« _4 L ^ 4 • ' 51 '

6 6X5 '6x5x4 6x5X4X3 ^^6x5x4x3x2 720*

The Probability that a (hall be in its place, and b^ r, d^ r,
/ out of their places, will be

-2 5_ 4. 'o «*> 4. J

•6 6x5 ^ 6x5x4 6x5x4x3 ^ 6x5x4x3x2

! 44_ ^- "

6x5x4x3x2x1 720* jSo *

The Probability that 4, ^, ^,i/, f, / (hall be all difplaced is
. ^^ ^ I. «.»/_« .^ 10 I ly ^^ 6

6 "6x5 6x5x4 **6X5X4X3 6x5x4x3x2

4. ! Qri,-^i4.-^ — -i- 4. '' .^ ' -

^ 6/.5X4X3X2X1 > * "• a 6 "^ 24 i2o

"• 710 720 141 "•

Hen^

i|i?|(IPi^|W|BiflPPJi*F^?»^*'«'T*''""''**'"'^'1'»«i*?^^

The Doctrine o/Chancbs. Isi

Hence it may be concluded that the Probability that one or
more of them will be found in their places is i — -i y^L^

k + -i^ — TiT = -^5 ^nd ^'^^^ ^hc Odds that one

or more of them will be fo found are as 91 to 53.

N. B. So many Terms of this lad Series are to be taken
as there are Units in n.

DEMONSTRATION.

TH E number of Chances for the Letter 4 to be In the
fird place contains the number of Chances, by which m
being in the firlt place, b may be in the fecond, or out of it:
This is an Axiom of common Scnfe, 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 4 to be in the firft place, there be Tub-
(traded the number of Chances that there are for 4 to 1)e in
the firft place, and b at the fame time in the fecond, there
will remain the number of Chances, by which 4 being in the
firfl place, b may be excluded the fccond.

For the fame reafon it follows, that if from the number
of Chances that there ate for a ^nd b to be refpedively in
the firft and fecond places, there be fubtrafled the number
of Chances by M'hich 4, b and c may be refpeftively in the
firfl, fecond and third places; there will remain the number
of Chances by which 4 being in the firft and b in the fe«
£ond5 € may be excluded the .third place : And fo of the
reft.

Let + a' denote the Probability that d fhall be in the
firft place, and let — a denote the Probability of its being
out of it. Likewife let the Probabilities that b fhall be in the
fecond place or out of it be refpeftively expreft by + 4"
and—*".

Let the Probability that, 4 being in the firft place, b fhall
be in the fecond, be expreft by 4'+ b"i Likewife let the Pro-
bability that a being in the firft place, b fhall be excluded the
fecond, be expreft by 4 — y.

Generdllj. Let the Probability there is, that as many as are
to be in their proper places, fhall be fo, and at the fame time
that as many others as are to be out of their proper places

R fball

■*^™i^

.1 w)M\i u} ip t^mif^m^mtf*tiw*\'mnmrmmwmimfmfmmmr9mmf^

• »■ *

62 TiBf Doctrine of Chances.

[hall be fo found, be denoted by the particular Probabilities
of their being in their proper places, or out of ihem, writtca '
all togiJthcr: So that for Inflance 4'+ T + r — /"'— e^
may denote the Probability that a^ b and c (hall be in their
prpper places, and that at the fame time both d and e (hall
be excluded their proper places.

Now to be able to derive a proper concIuHon by vertuc of
this Notation, it is to be obferved, that of the Quantities
which are here confidered, thofe from which the Subtraflion
is to be made, are indifferently compofed of any number of
Terms connected by + and — ; the Quantities which arc to
be fubtrailcd do exceed by one Term thofe from which the
fubtraction is to be made ; the reft of the Terms being alike
and their (igns alike: And the remainder will contain all the
Quantities that arc ahlce with their own (igns^ and alfo the
Qiiantity Exceeding, but with its (ign varied.

It having been demonftrated in what we have faid of
Permutations and Combinations, that / == -i-^ / J^ IT.

= — ^ — , 4' + *^ + /''= 1 , let-C _i_&c.

uxif I ' ' ' axil— im — a' 11 ' •(■—I mw

be rcfpedively called r, /, r , v &c. This being fuppofed^ we
may come to the following condufions.

TherefoK V — a ' = r — «

"iF^P = $ for the fame reafon that y* **=r •
if" -^ I/' + a' sz t

2» Thetef. ^^^+y^— g^= *- t

tf"—a' =s r— « By the fiift Condofioa

^"~-a'^V'= t—t B/ the ii,

3» There£ if"—a'— l/' = r-'i*-\-t
J*// + j/// + y' + a' = w

4»Theref. f'" ^ tT'-i- l/'—y = t—v

S"' + d"— m' = « — » By the aiT. Conclufioii
gn' jf. c^^— »'Jc V - t—v By the tfh.

S! Thcref. X'" ^ e"— g^ - y^ =s «-2 1 + p

"xT i u ii ir- ' r* "' '**' " M -' - '^ "' II ■ ■ ■"'^' " *-- -'^ -^'-^'^

Be Doctrine (/Chances. S^

tm^l/t^j s= r— 2 . + t By tlie %i. Cone
tf'Theref. H'" — V — J — d" = r— ? »4-3 t—p

By the fame procefs, if no Letter be particularly afligned
to be in its place, tlie Probability that luch of them as are
afTigned may be out of their places will likewife be found
thus.

— fl' = 1 — r For -V tf'and— a'togetbermake

7«Theref. —«' — *"= i ~2r+s

— a'— V = I — 2 r H- s By the 7A Conc;^.

—a'— U'^d '^ r— 2*4t By the ji. Cone* .

8« Theref. — a'— V' — d' = 1 -. 3 r— 3 1 — t

Now examining carefully all the foregoing Conclufions, it -
will be perceived, that when the Qaeftion runs barely upon
the difplacing any given number of Letters without requiring
that any other (nould be in its place, but leaving it wholly
indifferent, then the vulgar Algebraic Quantities which lie
on the right hand of the Equations, begin conftantly with
Unity : It willalfo be perceived, that when one finale Letter
is afTigned to be in its place, then thofe Quantities begin with r; .
and that when two Letter;^ are afligned to be in their places,
they begin with /, and fo on. Moreover *tis obvious, thac
thele Quantities change their (igns alternately, and that the
Numerical Coefficients which are prefixt to them are thofe*
of a Binomial Power, whofe Index is equal to the number
of Letters which are to be difplaced.

PROBLEM XXVL

AN T ghef9 number of difftrent Letters a, b, c, d, c, f c^ck
being edch of them npedted a cert m a number of times 9 dnd
tdken fromijcuoujlj as tt Haffens : To find the Probabititj tbdt of
fome of thofe Sorts^ fome one Utter of each mdy be found in its
f roper flace^ and at the fame time that of fome other Sorts^ no onr
Letter be found in its place^

SOLU-

jy w y»|LP i |i ; ij| i ji. ii ) i . pi .., li.jj^jlii W^w^i»"'w g i ^jM| i .t il ■"" n w«wwWHP*»' I « ^>»^i^— <«^1^;w<WWp^i"^»*^P>^

m^^ ^NvMiriMa^kM *^mt^^.mmm^tmtmm^ti^am^ ^ ** •«*<»— a^

64 The DocTKiuE of Cuhncis.

SOLUTION.

OUppofe n be the number of all the Letters, / the number
O of times that each Letter is repeated, and confequcmly
-i the number of Sorts : Suppofe alfo that p be the num-
ber of Sorts that arc to have one Letter of each in its place;
and q tlic number of SDrts of which no one Letter is to be
found in its place. Let now the prcfcriptions given in the
preceding Problem be followed in all rcfpefts, faving that r
muft here be made = -i-^ s = — '-!- — , t = '- . &a

and the Solution of any particular Cafe of tlie Problem will
be obtained.

Thus if it were required to find the Probability that no Let*
tcr of any fort fliall be in its place, the Probability thereof
would be

But in this particular Cafe q would be equal to J^'^
wherefore the foregoing Series might be changed into thi%
v/«. ___.

&a

CoroUsry L From hence it follows, that the Probability that
one or more Letters indeterminately taken may be in their
places will be .

* % »— I "T 6 ^ •— ix»— a 34 ^ » — ix» — ax»— J

CorolUrj 11. The Probability that two or more Letters
indeterminately taken may be in their places will be expreft
as follows,

CcrolliTi IIL The Probability that three or more Letters
indeterminately, la^en may be in their places will be as fol*
Ipws^

I ■■■

The Doctrine if Chancbs. €%

Corollary IV, The Probability that four or more Letters,
indeterminately taken^ may be in their places will be thus
cxpreft,

f
24

3«7

-I »—

a » —

c&c.

■—4

"'--7

The Law of the continuation of thcfe Series being nuni-
fed, it will be eafy to reduce them all to one general Sc«
ries.

From wlut we have faid it follows, that in a common Pack
of ;2 Cards, the Probability chat one of the four Aces may
be in the fiiil place ; one ot the four Duces in the fecood ;
or one of the tour Traes in the third ; or that fome one of
any other fort may be in its place (making 1 1 different places
in all ) will be expreft by the Series exhibited in the firft
Corolla ry.

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 edimate the rank which the Cards of the
fame Suite and Name are to obtain in the other; the Pro-
bability 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 expreft by the Series belonging to the firft CoroV
lary, making /r = ;x and / =: i : Which Series will in this

Cafe be I ^ + -i- J- + -i -—^ &a whereof

.2*6 24 T^ lao 7iO

jx Terms ought to be taken.

If the Terms of the foregoing Series are joined by couples,
the Series will become,
_i_ I _i t « I ! I » .

2 "• 1x4 "• 2x3x4x6 ^ 2x3X4x5x6x8 ^^ 2X3ii4X5X6x7Xlxio

SfC of which x6 Terms ought to be taken.

But by reafon of the great Convergency of the aforefaid
ileries, a few of its Terms will give a fumcient approxima-

S tioa

■>w<^^w^«i»pi» ■■ ■i<>»ai«w»i^wflW**^«"'«wiWP»wwiBWWW^riiy*""<WI*^'*^^**y*'^'^^^^'^'^'?'^*^^'*^^'^^^^

• .«p^v«. ^ rf^rfiM«aiiikM*i^B'i^»^"^MM*^B^w «^to* v<kM • «> »« m ■— I O ^ M ■ ■ > I < i^«i

66 T^he DOCTRINB of CHANCei

tion ia all Cafes required ; as appears by die following Ope*

ratiooi

-L. }st O.COOOOO
-I— sss 0.1x5000.

. ! — — = o«oo6944-i*^

S --5— =S 0.0001 744-;

' — B- — =s aoooooi-i**
Sum = 0. 6321x9 +

Wlicrefore the Frobabilicy that one or more like Oirds in
two diiTerent Packs may obtain the fame Podtion^ will be in
all Cafes very near a6}x; and the Odds that this will Hap*
pen once or oftner, as 63X to 3681 or as ix to 7 very
near.

But the Odds that two or more like Cards in two differ
rent Packs will not obtain the fame Fofitioni are very nearly
as 736 to 264 or 14 to $.

Corollary V. If J and B, each holding a Pack of Card%
pull them out at the fame time one after another, on con-
dition that every time two like Cards are pulled out, A fliall
give fi a Guinea; and it were required to find what confi«
deration B ought to give J to Play on thofe terms : The
Anfwcr will be, One Guinea, let the number of Cards be what
it will.

CoroUdry VI. If the number of Packs be given, the Proba^
bility that any given number of circumftances may Happen
in them all, or in any of tliem, will be found eanly by our
method. Thus, if the number of the Packs be i, the Proba-
bility that one Card or more of the fame Sute and Name, in
every one of the Packs, may be in the fame. Folition, will be
expreft as follows.

PR a-

^j w jm i 'JiJ ' ' .'- ■' ' L I

\

The Doctrine o/Chancks; €f.

PROBLEM XXVn.

IF A affd B ylyf together^ each with d certain number cf Bmts '
= n : What are their reffeSive Probabilities cf winnings
fupfofing that each of them want a certain number of Games of
being ufi

SOLUTION.

F/r/?, the Probability that fome Bowl of A* may be nea*
rer the Jack than any Bowl of ^ is -^^

Secondly^ Suppofing one of his Bowls nearer the Tack tbaa
any Bowl of A^ the number of his remaining Bowls is 0— 1»
and the number of all the Bowls remaining between them
is X0— i: Wherefore the Probability that fome other of
his Bowls may be nearer the Jack than any Bowl of A will
be "2't» ^^^^ whence it follows, that the Probability of his
winning two Bowls or more is -i-:

a«- I

Thirdly^ Suppofmg two of his Bowls nearer the Jack than -
any Bowl of A^ the Probability that fome other of bis
Bowls may be nearer the Jack than any Bowl of A will
be -^frn Wherefore the Probability of winning three
Bdwls or more is -i- x ',"L\ x -Jifrr • ^^^ continuatioo ^
of which procefs is manifelK

Fourthly^ The Probability that one fingle Bowl of B OialL '
be nearer the Jack than any Bowl of A is -i — 4-x *"' ^
or -j-x-^/, ,■ ; For, if from the Probability that one or

more of his Bowls may be nearer the Jack than any'Bowl |
of Af there be fubtra£led the Probability that, two- or more
may be nearer, there remains the Probability of one (ingle «,
Bowl of B being nearer : In this Cafe ^ is faid to Win ^
one Bo'wl 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 10 be -i- x T". x — - — . in which Cafe B is laid
ta win two Bowls at an End.

Sixtfdy^^

""'"""•■' nj ii Lii. I ■ ■ . i |wii^ i)i uuiyr yn>wffi»wfw^^w«yw||i 11 . 1 ^^ 1 LI ..111

^8 The Doctrine tf Chances.

Sixthly^ The Probability that B may win three Bowls at
an End will be found to be -i->x ""' ■x -*'\ x — =— .

The proccfs whereof is manifeft.

The Reader may obfcrvc, tliat the foregoing ExprefTions
might be reduced to fewer Terms; but leaving theni
unreduced, the Law of the proccfs is thereby made more
confpicuous.

Let it carefully be obfcrv'd, when we mention henceforth
the Probability of winning two Bowls, that the Senfe of it
ought to be extended to two Bowls or more; and that when
we mention 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, Firfly that -^ wants
one Game of being up, and B two; and let it berequired,
in that circumftance, to determine their Probabilities' ot win-
ning.

Let the whole Stake between them be fuppofed =s i. Then
cither A may win a Bowl, or B win one Bowl at an Endj,
or B may win two Bowls.

In the firft Cafe B lofes his Expeflatton.

In the iecond Cafe he becomes intitled to-^ of tt.e Stake.

But the Probability of this Cafe is -^ x — —: — : wherefore his
Expeflation arifing from that part of the Stake he will be
intitled to, if thi^ Cafe fliould Happen, and from the Proba-
bility of its Happening, will be -i— x • " -,

In the third Cafe B wins the whole Stake i. But the Pro-
bability of this Cafe is -i- x -—r: • wherefore the Expefta-
tion of B upon that account is — ^ x "" '

From this it follows that the whole Expc£lation of B is

-h ^ TTTT + 4- * ir^ or -^n "—, or -^i^^; which

2JI-1

being fiibtrafted from Unity, the remainder will be the Ex-
peftation of A^ viz,, -^j—. It may therefore be concluded,

that the Probabilities which A and B have of winning are
refpeftively as 5 ;» — x to 3 ;f — %.

*Tis remarkable, that the fewer the Bowls are, the grea-
ter is the proportion of the Odds j for if A and B play with

fingle

t^ii.i ■■ ii'i Ml I ■ .. 1. f «* ^ ■■ *!" ' *" *

.^_ I I I I ■ ■ ■ ■■ — ■ ■!■■

-• - ■^_.^. ..

The Doctrine o/Chances. 6^

fingle JBowls, the proportion will be as 3 to i ; if ihcy plajr
with two Bowls each, the proportion will be as 2 to i ; if
with three Bowls each, the proportion will be as 1} to 7:
yet let the number of Bowls be never fo great, tliat proportion
will not dcfcencl fo low as 5 to j.

Secondly^ Suppofe A wants one Game of bein^ up, and B
three ; then either A may win a Bowl, or B wm one Bowl
at an End, or two Bowls at an End, or three Bowls.

In the firft Cafe B lofes his ExpeSation.

If the fecond Cafe Happens, then B will be in the circum-
ftance of wanting but two to A^^s one ; in which Cafe his
ExpeQation will be l\Z^ ^ ^^ ^^ ^^^ ^^^^ before determined:
but the Probability that this Cafe may Happen is -J- % -j-i-. j
wherefore the Expeftation of B^ arifing from the profpefl of
this Cafe, will be-i-x — - — x g"'^ .

If the third Cafe. Happen, then B will be intitled to
one half of the Stake : but the Probability of its Happening
is -i-y ■-' X — - — ; wherefore the ExpeQation of B ari-

fmg from the Profpeft of this Cafe is -^ %

"'Vx

« — 4 '^ an— i >

2« - I

If the fourth Cafe Happen, then B wins the whole Stake i :
but the Probability of its Happening is -i-x ,^jj^ ^-ff-t

or .-i— X -—rr* wherefore the Expectation of B arifing from
the profpeft of this Cafe will be found to be -i- x /,V, •

From this it follows, that the whole Expedation of B
will be 9»>»-7iv;|^4 . ^^hjch being fubtrafled from Unity,

the remainder will be the Expeflation of-rf, viz. ^^" "'^' j^^

It may therefore be concluded, that the Probabilities which A
and B have of winning are rcfpedivcly as x^nn — 1 9 j9 + 4
to ^nn — 1 J » + 4.

N. B. If A and B play only with One Bowl each, the
Expectation of B deduced from the foregoing Theorem
would be found = o. which we know from other principles
ought to be =-4-. The reafon of which is that the Caic of

winning Two Bowls at an End, and the Cafe of winning

T Three

■' ' ■' ■■"" < ^1 .<pun^i»iai^i ■^rHp9HgFr"*P«^*««'Mi««BP««q^m«VfP«map«Mt<

'70 The Doctrine of Chances.

Three Bowls at an End, enter this conclufion, which Cafes
do not belong to the fuppofition of playing with fingle Bowls :
wherefore excluding thofc two Cafes, the Expcftation of B

will be found to be -L- X -—^ X - \IZ\ ' =^-7-, which will

appear if n be made = i. Yet the Expcftation of 5, in the
Cafe of two Bow's, would be rightly determined, tho' the
Cafe of winning Three Bowls at an End enters it: The rea*
fon of which i^, that the ProbabiUty of winning Tlirce Bowls
at an End is =, -?- x ""^ , which in the Cafe of Two

Bowls becomes = o, fo that the general Expreflion is not
thereby difturbed.

After what we have faid, it will be eafy to extend this
way of Reafoning to any circumftance of Games wanting
between A and B ; by making the Solution of each fimpler
Cafe fubfcrvicnt to the Solution of that which is immediately
more compounds

Having given formerly the Solution of this Problem, pro-
pofcd to me by the Honourable Frances RohrtSj in the Philo^
fophiedl Tranfactiofjs Number 339 ; I there faid, by way of
Corollary, that if the proportion of Skill in the Gamefters
were given, the Problem might alfo be Solved ; fince which
time Mr de Monmort^ in the fccond Edition of a Book by
himPublifhed upon the fubjcflof Chance, has thought it wortri
his while to Solve this Problem as it is extended to the con-
fideration of the Skill, and to carry his Solution to a very
great number of Cafes, giving alfo a Method by which it
may flill be carried farther: I very willingly adcnowledge
hb Solution to be extreamly good, and own that he has in
this, as well as in a great many other things, Ihewn himfelf
entirely mafter of the dodrine of Combinations, which he
has employed with very great Induftry and Sagacity.

The Solution of this Problem, as it is retrained to an e-
quality of Skill, was in my Specimen deduced from the Me-
t'lod of Combinations ; but the Solution wiiich is given of
ir in this place, is deduced from a Principle which has more of
fimplicity in it, being that by the help of which I have Dc-
monftrated the Doftrine of Permutations and Combinations:
Wherefore to make it as familiar as poHible, and to fhew
its vail extent, I ihall now apply it to the general Solution

of

PHILIP liiiiM"i.in"iji .

^ypipujiii^^iP^ljf^wwpp— jn^^^H— ip«iii> ■■! iiii ■ u » n iwB i B n ^^>^^ii>^^iwi . i iMfii * m

The Doctrine o/Chancei 71

of this Problem) taking ia the connderatioa of the Skin of the
Gamefters.

But before I proceed I think it neceflTary to define what
I call Skill : viz. That it is the proportion ot Chances which
the Gameders may be fuppofed to have for winning a fin«
gle Game with one Bowl each.

PROBLEM XXVIII.

IF A and B, tphofe proportion of SUB is as a to b, pljff fo^
gethery each with a certain number of Bowls : What an their
refpeilive Probabilities of winnings f^^fpopng each of them to want
a certain number of Games of being up f

SOLUTION.

Flrfij The Chance of B for winning one fingle Bowl be
ing b^ and the number of his Bowls being n^ it foDows
that the fum of all his Chances is nbi and for the fame rea*
fon the fum of all the Chances of ^^is na: wherefore the
fum of all the Chances for winning one Bowl or more is
na + nb'^ which for brevity fake we may call /. From
whence it follows, that the Probability which B has of wio*
nins one Bowl or more is -si^

Secondly^ Suppofin^ one of his Bowls nearer the Jack than
any of the Bowls ot A^ the number of his remaining Chan-
ces is 7f— 1x3; and the number of Chances remainmg be*
twecn them is / — b: wherefore the Probability that fomc
other of his Bowls may be nearer the Jack than any Bowl

of A will be ■ ''""''^/ : From whence it follows, that the Pro-
bability of his winning Two Bowls or more is — x *'^^^^ -

Thirdljy Suppofing Two of his Bowls nearer the Jack than
any o f the Bowls of A^ the number of his remaining Chan-
ces is w^^ x5 ; and the number of Chances remaining be-
tween them is i — x^: wherefore the Probability that (bmc
other of his Bowls may be nearer the Jack than any Bowl

of A will be -^ ^2^^ • From whence it follows, that the.

Pfoba-

wmmtnm

'I'lf^ *'iwi m ^w^wrMii g II ^ pwwwHWiiwwwM^»?Nwww>^w>

72 The Doctrine 0/ Chances.

Probability of his winning Three Bowls or more b -2i. x
n-ixh ^ jLiLil*.. the continuation of which procefs is

manifeft.

Fourth/j, If from the Probability which B has of winning
One Bowl or more, there be fubtrafted the Probability which
he has of winning Two or more, there w ill remain the Pro-
bability of his winning One Bowl at an End : Which there-

ni «^ ^ »— XX* _ nt _ / — If*

fore will be found to be -^ ^ x * ^ J, ^ - or

X

mh .. mm

or -=— X

Fifthly^ For the fame reafon as above, the Probability which
B has of winning Two Bowls at au End will be found to

be-i^x --'^^ - "

Sixthly^ And for the fame rcafon likewife, the Probability
^vbicb B has of winning Three Bowls ac an End will be

found to be JLLy^JEL^ ^ J^l^ ^T^ The con-
tinuation of which procefs is manifeft.

N. B. The fame Expeftations which denote the Probabi-
lity of any circumftance of 5, will denote likewife the Ro-
bability of the like circumftance of A^ only changing^ into 4
^nd d mto k

Thefe Things being prcmifed, Suppofe F/rfi^ that each of
them wants one Game of being up ; ^cis plain that theExpcda-

tions of ^ and B are refpeftively -^ and J^^ Let this Ex-
pectation of B be called P.

SecojfJfy^ Suppofe J wants One Game of being up and B
Two, and let the Expeftation of B be required : Then either
j1 may win a Bowl, or B win One Bowl at an End, or B
win Two Bowls.

If the firft Cafe Happens, B lofes his Expeftation. .

If the fecond Happens, he gets the Expeftation P ; but tlic

Probability of this Cafe is -2i- x ~^: wherefore the Ex-
pe£latioo of B arifing from the poflibility tliat it m^y fo Hap-
pen is -si-x-j^x.p.

If

yjiHiA^uHii I ii ji ii y ii .iM ii . i .11 j, i i ■ ■■■i .i.uju|ny i . , ^.M , |,j i Jii i i i M) ii .ni i ■ ■ M.„j.. ,, j , „ ..^

\

\
\

The Doctrine 0/ Chances. 73

If the third Cafe Happens, he gets the whole Stake i ; but
the Probability of tl)is Cafe is J±. x -J-fi-, wherefore the
Expeflation of B arifing from the Probability of this Cafe
is -T->^-7^x I.

From which it follows that the whole Expectation of B
will be ^x-~!V P + -T^)c-~£x* Let this Expeaation
be called j^

Thirdly^ Suppofc J 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 Expcftaiion J^; or Two Bowls at an End,
in which Cafe he gets the Expcdation P ; or Three Bowk
in which Cafe he gets the whole Stake i. Wherefore the

Expectation of B will be found to be —- x '^:^ x Q^

Aa infinite number of thefe Theorems may be formed in
the fame manner, which may be continued by infpeAion, ha-
ving well obferved how each of them is deduced from the
precedinjg.

If the number of Bowls were unequal, fo that J had m
Bowls and B n Bowls ; Suppofing wn + »i = /, other Theo-
rems might be found to anfwer that inequality : And if that
inequality Ihould not be conftant, but vary at pleafure ; c*
ther Theorems might alfo be formed to anfwer that Varia-
tion of inequality, by following the fa(ne way of arguing.
And if Three or more Gamefters were to play together un*
der any circumflance of Games wanting, and of anjr given
proportion of Skill, their Probabilities of winning might be

determined after die fame manner;

• • • J

PROBLEM XXIX.

TO find the Expeltdtioft of A when with 4 Die cf awrjf gi.
vttt number of Fdces he undertdhes to flhtg dnj .dettrmi^
jfdte number of them in dnj given number of Cdflj.

U SOLU-

' ■ ^' 1> Wtl l U lW||i^W<%.pw^Hp»

IPiV99Mn*WVi"Vi««p4P"«>'^iP^Wll9i"^^

74 Tfir DocTniNE 0/ Chances,

SOLUTION.

LET f + 1 be the number of all the Faces in the Dte^
n the number of Cafts, / the number of Faces which
he undertakes to fling.

The number of Chances for an Ac e to come up once or
more in any number of Cafts ir, is p+ iV — p ■ : As has-
been proved in the IntroduQion.

Let the Duce, by thought^ be expunged out of the Die, and
thereby the number of its Faces reduced to p^ then the num--
her ot Chances for the A ce to come up will at the fame time
be reduced to ^— pH?". Let now the Ducc be reftored^
and the number of Chances for the Ace to come up without
the Duce, win be the lame as if the Duce were expunged.
But if from the number of Chance s for the Ace to come up
with or without the Duce^ wjc from p+T" — / "^ be fubtraded
the number of C hance s for the Ace to come up without the.
Duce, via^ f — p— ii% there wiD remain the number of
Chances for the Ace and Duce to come up once or more^
which confequently will be ^+T* — ixp* + p— i^".

By the fame way of arguing it will oe proved, that the
number of* Chances for the Ace and Du ce t o come up^
without the Trae will be p^— ixpl^'+p'^*, and con-
fequently, that the number of Chances for the Ace, the Duce
and Tra e to c ome up on ce or more, will b e the diff erenc e
between p-fi'" — x xf>'+ p— ii* and p* — xxp-i* *Vp— 2^%
which therefore is p-p?" — 3 x p» + 3 5« p— r* + p— ?".

Again it may be proved that the number of Chances for
the A ce, the Duce, the Trae a nd Q uat cr to come up, is>
p+i" — 4xp"+6xp— !.■ — 4xp— 2^» + p— 3-»i the conti-^
nuation of which Procefs is man ileft.

Wherefore if all the Powers p+i^,^,p— i'",p— 2'",p— g**
&C. with the Signs alternately Pofitive and Negative, be
written in Order, and to thofe Powers there be prenxt the re*
(pe6live Coefficients of a Binomial raifed to the Power A the
fum of all thofe Terms will be the Numerator of t n e Ex.
peAation of ^, of which the Denominator will be p+ il*.

EXAM-

■ _•-•-■■-■■

Tife Doctrine qfCriANCBi 7^

EXAMPLE L

LET six be the number of Faces in the Die, and lei'
A undertake in Eight Cafts to fling both ao Ace and

a Duce : Then his Expeflatioa wiU be g'-^^M'-t-t* .

55 jXif" . = ju nearly^

EXAMPLE n.

IF A undertake with a common Die to fling all the
ParM in t «. Cafts. his Expcflation will he found tn hi>

<•: — n

races in i% v«<tii9y iin c&jtiv.iciuuu wiu u«

nearly.

EXAMPLE IIL

IF A With a Die of 36 Faces undertake to fling two given
Faces in 4} 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, nis E^

peaation will be ?6^'---M5^ + H"' ^ .^ ncariy.

N. B* The parts of which thefe Expectations are com*
pounded, are eafily obtained by the help of a Table of Lo*
garithms.

PROBLEM XXX-

1^0 fisfd in haw mAnj Triah it mi be frotdUe that A vritb
d Dii of dnj given number of Pdces fbdO throm- dnj
frofojed number of them.

SOLUTION.

ET ^+1 be the number of Faces in the Die, andY
the number of Faces which are to be thrown. Dividle

the Logarithm of 7^ by the. Logarithm of.i±L, andi

thcj

L

»■ < l- l 11 1 i^ W f lMt»>»-^>«^«MyilWWBMB|»^M^;p^p^||fl^M|i^JpByW^p^ggi»»ayP^W»WffW*'^f1!^*W"W^^^

WWi*

76 The Doctrine cf Chances.

the Quotient will exprefs Dearly the number of Trials requu
fite, to make it as probable that the propofed Faces may be
tlirown as not.

DEMONSTRATION.

SUppofe Six to be the number of Faces which are to be
thrown, and n the number of Trials: Then by what has
been dcmonft rated in the preceding Problem, the Expcfla*
tion of A will be,

Let it be fuppofed that the Terms p+i, f^p — i,/— x&c.
are in Geometric Progreflion ( which luppofition will
very little err from the truth, cfpecially if the propor-
tion of/ to I be not very fmall ). Let now r be writ.

ten inftead of - ^"^' > and then the Expectation of A will be
changed into I —-;r+ T^r — -7V + -7V
or J — -nr * ^"^ ^^^^ Expectation of A ought to be made
equal to -i-, (ince by fuppodtion he has an equal Chance to
win or lofe : Hence will arifc the Equation 1 — -^* = -y

orr*» T", from which it may be concluded that
n x Log. r, or ;f X Log. Ji+L s= Log. T", and coofequent-
ly that n is equal to the Logarithm of T" divided by the

Logarithm of -y- = -^i^ . And the lame Demonftration wiD
hold in any other Cale.

EXAMPLE L

TO find in how many Trials A may with equal Chance
..undertake to throw all the Faces <h a common Die;

The

Sff^W^''^'**"— ^'^ ' ■ ■ ■ ■■ '<' ' ■■■i . i M " ■■ tf i J. a ui^tn i f. i m. iii j BB ■ nm M ^ i n i.i i , .^ ,y

*^ • ^Hl^a^^B^v. ^a

7%e Doctrine o/Chancei 77

The Logarithm of "T' =s 0.9 6x 1 7 ; 3 ; the Loga-
rithm of f'^^ or -|— = 0.0 7 9 1 8 1 1 : Wherefore m
== ''•^f^!!!^ = I X +. From hence it may be concluded

0.079 1 1 s ' ^

that in 12 Cads A has the worft of the Lay, and in 13
the bed of it.

EXAMPLE IL

TO find in how many Trials, J may with equal Chance^
with a Die of Thirty-fix Faces, undertake to throw
Six determinate Faces ; or, in how many Trials he may with
a Fair of common Dice undertake to throw all the Dou-
blets.

The Logarithm of T" being o. 9^11753, and the

Logarithm of ^"^' or -^ being 0.0 1 21345'; it follows

that the number of Calls requifite to that effeft is ""-p^^ns^

or 79 nearly.

But if ic 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 (liould go for no-
thing ; then the throwing of the Six Doublets would be like
the throwing of the two Aces Six times; to pioduce which
efTtft the number of Cafts requifite would be found by Mul-
tiplying 35 by y.668, as appears from our VII/A. Problem,

and confcquently would be about 198.

I f

/f. B. theFraftion T" may be reduced to j ^^ —

I— Vt ' /; .

which will Facilitate the taking of its Logarithm.

PROBLEM XXXI.

IF A, B, C PUj together on the foBowmg conditions ; Fir ft ^ thdt
ihtj (bad each of them Stake i I. Seco/sdlj^ that A and BJtdt
tegin the PUy \ Thirdly^ that the Lojer (ball yield his fUce to the
third Man^ which is to be obferved confiantly aftertpards ; Fourthly^

X that

"»i»w^jn Mil— I . I u iMiit^ ^w i i p ■wMiii 'F W'^/r9tmmmi^t^ir9'i00mm^mmmf'f'''^i''''''^^ ■ wjmi jn wqtgy.gsa;

7$ The Doctrine of Chancbi

tbai the Lofer (bsB be fnei d certain Sum p, which it to
ferve to incresfe the common Stock ; Laftlj^ that he /ball Win the
whole Sum defofited at fir ft ^ and increafed by the fever al Fines ^
who (hall firjl beat the other two juccfffivelj : ^Tis demandedt
what is the Advantage of A and B, wfwm we fuffofe to^ begin'
the ?lg.

SOLUTION-

EC fi^ fignlfie that B beats ^, and ^ C that^ beats C;
and let always the firft Letter denote the Wionery and
the fecond the Lofer.

Let us fuppofe that B beats A the firft time : Then let us
inquire what the Probability is that the Set (hall be ended
in any given number of Games ; and alfo what is the Piioba-^
bility which each Gamefter has of winning the Set in that
given number of Game&

Firfi^ If the Set be ended in two Games, fi muft neceflk«
rily be the winner ; for by Hypothefis he wins the firft time :
Which may be exprelTed as tollows.

L

il

BA
BC

Seeoudljf If the Set be ended in Three Game% C muft be
(he winaer ; as appears by the fblldwing Scheme.

I I BJ
%\ Cfi
3 I CA

Thirdfyf If the Set be ended in Four Game^ A mnft be
the winner ; as appears by this Scheme.

I
ft

3

4

BA

ca

AC
AB

Fourthly^ If the Set be ended in Five GaaieS| B muft be
the winner ; which is thus cxpreiTcd,

i] BA

Wfmfm^ii^ ' • ^ i - ' .. ■ "^ 'S* . ^' ^ ' ". ' *., ' '^ ' "-- ' ■ w ' - wi'*«*w!i!— ^1 jm^m w u.i

-h*.

The Doctrine o/Chancb&

79

I

BA

%

tt

\

AC

4

BA

5

BC

Fifthly^ If the Set be ended in Six Games, C muft t>e the
winner} as will appear by ftill following the lanic VtxKJ^
thus.

I'

ISA

3

CB

3

AC

4

BA

S

CB

6

CA

And this Procefs recurring continually in the lame OHer
needs not be profecuted any farther.

Now the Probability that the firfl: Scheme (hall take place
is -i-y in conlequence of the fuppofition that £ beats ^ the

firft time} it being an equal Chance whether fi beat C^ or
C beat A.

And the Probability that the fecond Scheme fliall take place
is -i- : For the Probability of C beating fi is -2^ and that

being fuppofed, the Probability of his beating A will alio
be -i- ; wherefore the Probability of B beating C> and then A^

wiu'be^-x^or -5-

And from the fame condderations the Probability that the
Third Scheme {ball take place is -^ : and fo oo.

Hence it will be eafie to compofe a Table of the Proba*
bilities which B, C, A have of winning the Set in any givea^
number of Games ; and alfo of their Ex]^eftations : Which
ExpeQations are the Probabilities of winnmg Muldplyed by
the Stock Three depofited at firft| and increafed fuobdffivdy
by the feveral Fines.

Table

M 1 1 11 M.JU ill . ilf. l j l tW I I. II >l |. lll "t ' . f ''«' ■ H. i| I JH I .lPW r»*^WWWWf^|||pH|j^

- ■ "•-* . !-•

8o

The DOCTRIKEO/CHANCB*.

Table of the FrobabUities, &C.

0<

B

c

A

o

• • • • ■

St

3
4
5

7
8

9

10

i ^3+Jf

-T-x 3+4^

*'♦ X 3+7/
5.2 x3+»o^

.« ^ 3+5/>

IT'* 3+«/'"

« A # • •

* V 9 w V •

. • • • • •

II

-rsT '<3+»V

Now the feveral £xpe£lations of B^C^ A may be fum-
med up by the foUowiag Lenuna*

LEMMA.

^^. JliifL+Ji+iiL+ I^ ^ JiJ^ ice Ai infinitum

is equal to ^-, + j;^* •
Lee the ExpcflatioDS of B be divided into two SerieSi viz^

1

+ ^ +

16
If

16

+
+

JJL
Its

+
+

1034

lip

1024

^ The (irft Series conAitutes a Geometric Progreflion con-
tinually decrcafing, whofe fum will be found to be -il-;

The fccond Series may be reduced to the form of the Se-
ries in l>ur Lemmai and may be thus expreft.

^x

n " mT n ■ ^^r^W'Biff<M*yiFy»^WWWWWi^)iB^pywypwFWi»^wiW!ii>^^ |i* ;;Hlii w iH. » l ^^

\

The Doctrine e/ Chances.

8i

dividing the whole by
(hall have the Series -A

II

14

a» -r -jT- &C. Wherefore
L-, and laying afide the Term x, wc

It

s> I s» + Ti^* ^^* which

has the fame rorm as the Series of the Lemmay and may be
compared with it : Let therefore n be made =5 jr» i a 3 and

* = 8, and the fum of this Series will be -i- + .1-, or

-iL. . to this adding the (irft Term t^ which had been laid
afide, the new fum will be "'^
plied by 4.

^^ • and that being Multi*
the Produft will be -ii-^, which is the fum

of the fecond Scries exprefling the Expeftations of B : From
hence it may be concluded, that all the ExpeQations of B
contained in both the abovemcntioned Series will be equal

And by the help of the foregoing Lemma it will be
found likewife that all the Expe6tations of C will be equal

It will alfo be found that all the Expectations of A will
be = -f + ^f.

Hitherto we have determined the fevcral ExpeSations of
the Gameftersi upon the fum by them depofited at firfl*, as
alfo upon the Fines by which the common Stock is increa*
fed : It remains now to Eftimate the feveral Risks of their
being Fined ; that is to fay, the fum of the Pro^bilities of
the'u* being Fined multiplyed by the refpeAive Quantities of
the Fine.

Now after the fuppofition made of J being beat the firft
time, by which he b obliged to lay down his Fine pj B and C
have an equal Chance of being Fined after the fecorul Game,
which makes the Risk of each to be = —^^ ^s appears
by the following Scheme.

BA
LB

oc -;

BA

Be

Til

■ . 1 ■ ; I m

'yfommm

n^mmrmmt

■pw^wi^p^p^nr

8a The DoctRiNE of Chances.

In the like manner, it will be found that both C and A
have one Chance in four for their being Fined after the
Third Game, and confcquently that the Risk of each is -i-^,

according to the following Scheme.

BA BA

or

CB
AC

CB
CA

And by the like Procefs it will be found that the Risk of
£ and C after the fourth Game is -^ /.

Hence it will be eafie to compofe the following Table which,
cxprelfes the Risks of each Gamefter.^

Table of risks.

B

T

3

4

5
6

7
8

9
iffc.

c

a
t

ir?

4

P

^P
irP

12»

256

296

In the Column belonging to B, if the vacant places were
filled up, and the Terms -5-/, — 7-/* "a^r^f ^^ ^^^^
Interpoledy the Sum of the Risks ofB would compofe one
uninterrupted Geometric Progre(fion, whofe Sum would be
=: p ; But the Terms interpoled conftitute a Geometric Pro-
greflion whofe Sum is = -|-^: Wherefore^ if from/ there

be fubtraQed -^/> there will remain -i- f for the Sum of

the Risks of B^

In like manner it will be found tl at the Sun of the Rbks
of C win be sa ^f. }i^ji^

lU ' J i II ^J^ i^p m

\

I

The Doctrine 0/ Chances. 83

And the Sum of the Risks of Ay after his being Fined the firft
time, will be = -i- f.

Now if from the feveral Expeftations of the Gamefters
there be fubtra£led each Man^s Stake, as alfo the Sum of bis
RiskSi there will remain the clear Gain or Lofs of each of
them.

Wherefore, from the Expeflations of B = -12- + -^p

Subtrafting firji his Stake = i

Then the Sum of his Risks = -1*^

There remains the clear Gain of 5 = -4- 4. -^^p

7 • ^9

Likewife, from the Expe£lations of C = -i- ^ »^ p

Sabtrafting firfi his Stake = i

Then the Sum of his Risks = JL,p

There remains the clear Gain of C =s — -i- 4. < a

In like manner, from the Expectation of ^ =s .5. ^ 2L,p

Subtrafting, Firfi, his Stake ss \ ^^^

Secondljf the Sum of bis Risks s Jup

Ldjlljy the Fine f due to the7 ^^
Stock by the Lofs of the firft Game S f

There remains the clear Gain of ^ =s — ♦ !f.i^

But we have fuppofed in the beginning of the Game that
A was beat ; whereas A had the fame Chance to beat BjZsB'
had to beat him: Wherefore dividing the Sum of the Gains
of 5 and A into two equal Farts, each part will be -i- — -JL* .

which confequently mud be reputed to be as the Gain of each
of them.

CorolUrj !• The Gain of C being — -i- + -^ ^, Let
that be made = o. Then p will be found = -i-. If there-
fore the Fine has the fame proportion to each Man^s Stake
as 7 hds to 6, the Gamefters play all upon equal Terms :^
But if the Fin^ bears a lefs proportion to the Stake than 7

to

M . X mm m mn m MM , , , i J imi ■! i JlJL ii J i H ip | IH >>A P » !"■ ^ | H II nm > ■ "^ ■» ■ ■ ! . ^ ■■. . ■■i ■ w 'Jl I ■»> ■WWJfj

84 7^^ Doctrine 0/ Chances.

to 6, C has the difad vantage : Tlius, Suppofing f ^ I9 his
Lofs would be -J — But if the Fine bears a greater pro-

portion to the Stake than 7 to 6, C lus the Advantage.

CorolUry IL It the Stake were conftant, that is, if there
were no Fines, then the Probabilities of winning would be
refpcflivcljr proportional to the Expectations ; wherefore fup-
poiing f = o, the Expcftations of the Gamcfters, or their Pro-
babilities of winning, will be as -i^, -i-, -i-, or, as 41 x, 1 :

But the increafe of the Stock caufes no alteration in the Pro*
babilities of winning, and confequently thofe Probabilities
are, in the Cafe of this Problem, as 4, i, i ; whereof the firft
belongs to B after his beating A the Hrft time ; the fecond
to C, and the third to A : Wherefore 'tis Five to Two, before
the Play begins, that either AorB wins the Set ; and Five
to Four that one of them, that fhall be fixt upon, wins ir.

CoroUary III. If the proportion of Skill between the Game-
Aers Aj BfC be as j, ^, c rcfpeflively, and that the refped-
ive Probabilities of winning, in any number of Games after
the firft, wherein B is Suppofed to beat Aj be denoted bv
jr, Bl'% B'^ B^ &c, L\ C^, 0^\ C^' &c. A\ A^\ A'^,
A^\ &C. ir will be found, by the bare infpeQioo of the
Schemes 1>elon£ing to the Solution of the foregoing Problem,
that

cr =

Let -—J x-^^ X ■—- be made = m; then it will

plainly appear that the feveral Probabilities of winning will
compofe each of them a Geometric Progreflion, for

TO*" ___

JD SS

^mft^y^ iBn.! w II ttuf^mw^t^^mmmmm^mmtrnmenyji jhhji.wi.ilji!h. wj umapm.

-k.__.

7%e DocTRiNBo/ Chances.

'»$

B^ = »B' 1

(f ■A m (f

iT =«. iT

Jg'" « m tC"

(T =mC

-rf' = «• ^

B' ^m BT

(f ^mC'"

.i*' =^ « A"

&C

&C

itCi

Hence a Table of EiLpe£tations and Risks may eafily be
formed as above ; and the reft of the Solution carried oo by
following exadly the fteps of the former.

When the Solution is brought to its concIuHon, it will be
neceflary to make an allowance for the fuppofition made thac
B beats A the firft time, ^vhich may be done thus.

Let P be the Gain of B, when expreft by the Qpantides
dj bj Cy and Q^ the Gain of Jy when expreft by the (ame :
Change 4 into b and b into 4, in the Quantity flj then the
Quantity refulting from this Change will be the Gain of B, ia
cafe he be fuppoled to lofe the firft Game. Let this Quan-
tity therefore be called A, and then the Gain of B^ to be efti*

mated before the Play begins, will be - ^^^jj.^^ ,

PROBLEM XXXIL

F Four Camejlers A, B, C, D PUj on the conditions of the
JL f^^^Sfi^'^g Problem J dnd be be to be repute J the Winner^ vfbo
jball best the other Three fuccefflvefy : What is the AivantMge ef
A And By whom we Suppoje to begin the PImji f

SOLUTION.

ET BJ denote, as in the preceding Problem, that B
beats Ay and AC that A beats C ; and generally let the
firft Letter always denote the Winner and the fecond the
Lofer.

Let it be Suppofcd alfo that B beats A the firft time : Thea
let it be inquired what is the Probability that the Play ihall
be ended in any given number of Games ; as alfo what is the
Probability which each Gamefter has of winning the Set in
that given number of Games.

Z /%/,

L

9>\ 9 — I J PWWW^W

^ . ^ m * . '

H. li UM l jIJ I M M l J»HH- ' ■ . ■ " .M »> '■'

n0mm

■^'^TT*^WW

2€ The Doctrine o/Chancbi

/vr/, If the Set be ended in Three Games, B mud necei^
farily be the winner : Since by Hypothefis he- beats A the
firft Ganae, which is exprefTed as follow^

1-
3

BC
BD

Seconitj^ If the Set be ended in Four Game^ C muft be
the winner ; as it thus appears.

I

X.

3

BA

CB.

CD

4l CA

ThirJfyt If the Set be ended in Five Gaines, D will be
the winner ; for which he has two Chances, as it appears by
the following Scheme.

I

X

3

4
S

BA

BA

i

CB

BC

DC

or

DB^

DA

DA

DB

DC

Fourthly If the Set be ended In Six Games, A will be :
the winner ', and he has three Chances ibr it, which are thus .
eoHeaed,

BA

BA

BA

CB

CB

BC

DC

CD

DB.

AD

AC

AD

AB

AB

AC

AC

AD

AB

I

3
4

S

6

Fifthljy If the Set be ended in Seven Games, then B will >
have three Chances to be the winneri and C wUl have two j
ihus^

i\BA.

t;. '-r •■

■■n

'W«n-'«wwni

m a ■ m i

*mmm

the Doctrine 0/ Chances.

»7

a

3
4

6
7

£>4 £i4 B^ B/4

CB

DC

AD

BA

'BC

BD

CB
DC
DA
BD
BC
BA

CB
CD
AC
BA
BD
BC

BC

DB

AD

CA

CB

CD

BA

BC

DB

DA

CD

CB

CA.

Sixthlj, If the Set be eDded in Eight Games, then D u'iO
have two Chances to be the Winoer, C will have thre^ and:
B alfo three, thus

X

a
3-

4

S
6

7
8

BA BA BA BA BA BA BA BA

CB
DC
AD
BA
CB
CD
CA

CB
DC
AD
AB
CA
CD
CB

CB
DC
DA
BD
CB
CA
CD

CB
CD
AC
BA
DB
DC
DA

CB
CD
AC
AB
DA
DC
DB

BC
DB
AD
CA
BC
BD
BA

BC
DB
AD
AC
BA
BD
BC

BC

DB

DA

CD

BC

BA

BD

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; in this manner,.

3
4^

5

6

7
&

9
so

&C.

I B
I C
1 D

3 B +iC

J C+ xD+ }B

J D + xA + >C+3D + t.A'

J A + iB + 3D+3A+ tB + 3A4-aC + }D

TIko

■«!' »li . • wvmmmmmvm

IBS rj&eT>ocTRiNE of Chances.

Then Examining the formation of thefe Letters, it will
appear ; Firji^ that the Letter B is always found (b many
times in any Rank, as the Letter A is found in the two pre-
ceding 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. Ihirdlj^ that D is found (b many times
in each 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 each, as D is found in the preceding Rank, and C
in the Rank before thac

From whence it may be concluded, that the Probability
which the Gamefter B has of winning the Set, in any gi-
ven number of Games, is ^^ of the Probability which A

has of winning it one Game fooner, together with — i- of the

Probability which A has of winning it .two Games foo-
ner.

The Probability which C has of winning the Set, in any
given number of Games, is -i- of the Probability which B

has of winning it one Game fooner, together with —L- of the

Probability which D has of winning it^ two Games fooner.

The Probability which D has of winning the Set, in any
given number of Games, is -i- of the Probability which C

has of winning it one Game fooner, and alfo -i- of the

Probability which B has of winning it two Games foo-
ner.

The Probability whith A has of winning the Set, in any
given nupfiber of Games, is -i- of the Probability which D
has of winning it one Game fooner, and alfo -L. of the

Probability which C has of winning it t\^'0 Games foo-
ner.

Thefe things being obfervcd, it will be eaGe to compofe
a Table of the Probabilities which B, C, D, A have of win-
ning the Set in any given number of Games } as alfo of their
Expcftations, which will be as follows.

A TA-

^ ^ _^^..^4jti^^iKl w ■'< I

The Doctrine 0/ Chances.

^9

Table

of the Probabilities, &C.

//

M
fM

V
X'

Vff

Vltf

fx

X

3
4
5

6

7

8

9
10

II

B

C D

A

♦ '<4+3?

-i-X4+4P

w '<4+5?

X4+^

-t-X4+71»
lk-X4+%»

H ><4+7P
;l8X4+8p

» — :

ji2 X4+ICJJ

■:i5-x4+»v

-nrx4+8p
,014X4+ lip

:::::: 1

4
2)6

51a

9

to34

X4+i<¥

-;j*rX4+»>P

12

^*^-^^v

1^x4+12/^

M4i.X4+iap

X4+IV

The Terms whereof each Column of this Table is com-
pofed, being nor eaflly fummable bv any of the known Me-
thodsy it will be convenient, in order to Hnd their Sums^ to
ufc the following AnsUyps.

Let B' + B"+ B"^+ B^" + B^+ B*' &c. rcprefent the
refpeftive Probabilities which B has of winning the Set, \a
any number of Games, anfwering to 3, 4, y, 6, 7, 8 &c;
and let the fum of the T'^-obabilities Ad infinitum be fuppo-

fed = ^

In the fame manner, let C + C" + C' ^ C ^ a j^
C &c reprefent the Probabilities which C has of winning,
which fuppofe = x..

Let the like Probabilities which D has of winning be rc-
prefented by 1/ + D" + D'" + U'" + D»^ + Z)*^ &c which

fuppofe = V.

Laftly, Let the Probabilities which A has of winning be
reprefented by il' + ^4^' + A"Jt A^'" + ^-^ + jf' &c. which

fuppofe = «.

Now from the Obfervations fet down before the Table

of Probabilities, it will follow, that

A a B'=

J, I ■ ■ I HI J ■ w mt I . j ii i p^^«tPw^^y^q^tpr

■ JJ IP J. IW J " '»■*"'" ' * ' ^ '

m*m imii^^ m >

^o The Doctrine of ChamceI

B' »

B'

B' =

BT

r =

I A" ■\-

f
4

A'

B"'^

-^A"-^

1

4

■ A"

B" =

; A"'-\--

4

A"'

B' =

4-^' +

1

4

A'"

&C.

From which Scheme we may deduce the Equation follow-
ingt J = -^ + -^- x: For the Sum of the Terms in the
firft Column is equal to the Sum of the Terms in the other
twa But the Sum of the Terms in the firft Column is j by
Hypothefis ; wherefore 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 fccond Co.
lumni I argue thus,

^/ + ^// + j^'' + A'''' + ii*" + -rf'^ is = jp by Hypoth.
or A''^A'''^A'''^'A^^A^\s =*— ^'

and ^^'^+^^''^+^^''''+4-^''+T^«=T*--r^'

Then adding B* + B" on both fides of the laft Equation,
we (hall have
BfJ^g'j^^A''^-^ A'" + -V A'^'Jt 4 A'^J^ ^A^icc.

2 3

But ^' = o, B' = — I fi'' = o, as appears from the Ta-
ble : Wherefore the Sum of the Terms of the fecond Column
is equal to -i- * + -J-.

The Sum of the Terms of the third Column is -^ x by

Hypothefis; and confequently the Sum of the Terms in the
fecond and third Columns is = -l- x -^ -i-. From whence

it follows that the Equation y ^ -i- + -^^ jp had been
rightly determined.

In

JMimW y ' II* ■ P J l. ' W P u I ■ Mt ' iJ WW I * n J. I III

le Doctrine 0/ Chances. 9\

the fame manner, if we write

aing like the former we (hall at length come at
>o likewife if we write

P" = -i- C'*' + _i_ fi*'

2>" = 4- C" + -T- fi"

educe the Equation v s -i « -(- ^ j.
if afcer the fame manner we write

A = A'
A" = A^
A" =4-1^ +-1-C

A ^jL.y'" + 4-(r

ibtain the Equation « s ^v 4- J. el

Now*

7r" ifMf j^nt^Vn^pfv*

'■ ■ ■ ■ *^^i9!!pmn^a^rPifri>;r*^^'<<l<l«*<«N!V*«^^

5?

The Doctrine of Chancer,

Now thefc Four Equations beiag refolved, it will be found
that

iJ' + B^ + B'" + B^'' + B^+ B^' &c = J =

149
32

149
25

149

Thcfe Values being once found, let b^ c^ i^ Mj which arc
commonly employed to denote known Quantities, be refpeft-
ively fubftituted in the room of them s to the end that the
Letters j', z^ v, x may now be employed to denote other
unknown Quantities.

Hitherto we have been determining the Probabilities of win-
ning : But in order to find the Expectations of the Gameflers,
each Term of the Series exprefling thefe Probabilities, is to
be multiplyed by the refpctlive Terms of the following Se-

ries; 4+?P, 4+4f» 4+5^ 4+^^ &c.

The fint part of each Product being no more than a
Multiplication by 4, the fums of all the firft parts of thofe
Products are only the fums of the Probabilities multiplied
by 4 ; and confec]uently are 4^, 4r, 4^ and 44 refpedive-

But to find the Sums of the other parts.

Let 3 B> + 4 ^> + S B!'f + 6 B'^'^f ice be =/ j,

3 C/> + 4 C> + 5 C> + 6 C'7 &c ^fz,

3 D> + 4 !>'> + f ly^'f + 6 !/"> &a . a /<w,

3 ^/ + 4^>+ 5^7 + 6^">&a s/jT,

Now Since 3 ^ »

jiJ'

4i^'»

4B"

f JT" = •

ri-^" +

"4 •

6 tl'" =

4-^"+ .

J^A'

yJB" =

: A"'\

8 £^ s.

-l-^V+'

&C.

4

Ic

pP>«^i«B«wn

i«i0lp*'W'|'^TM^iiff«9«i^nMi

«BvnaM>

«wt

The Doctrine 0/ Changes. ^3

It follows, that jr =s -i- + -1- X + «. For the iirft Co-
lumn is s= y^ by HffotlKjis.

Again, 3 ^ + 4 ^' + j /4"' + « A"" + 7 if &c. = *by /^

Pothefs.
But i4' + ^" + ^"'+ A"''-\- A^Sic, has been found = *,
Wherefore adding thcfe two Equations together, we (hall

have 4 a:+s ^"+ 6 ^4'"+ 7 ^""+ 8 ><''&<;=* + «.

or ±.At+ 4.^'+ 4^"'+ 4. ^""+ 4.^>'8fc = 4.*+-f«.

Now the Terms of this laft Scries, together with j B' +'
4 B'^y compofe the fecond Column : But 36'= .2., and

4 B'^ = o, as appears from the Tabic. Confequently the
lum of the Terms of the fecond Column ts s .1. 4. .i— x

r 2

By the fame Method of proceding, it will be found, that
the fum of the Terms of the third Column is = -!- x

From whence it follows that jf = -L+ Xx + -ij + JLX'\'J.g^
0^-^=4- + ^^ + ^-

In the fame manner if we write

3 c « 3 c

8 C' = -f- 5^ + -f IT

We (hall from thence deduce the Equation £= J~j j^ _l^

B b So

^ m*- m mtm ^dte.^

^4 The Doctrine 0/ Chances,

So likcwifc in the fjine inanoer, if we write

3 D' = 3 1/ _

Laftly, if after the fame manner we write

3 A' = 3 A'

4 ^'' = 4 A"

<f ^'" = -f- !>"' +-fC''

We fhall deduce the tu'o following Equations, viz.
V = J_a + 4-c + J_ » + ^*. And x = -i-v+-L<<+ -L»

■T^ a

Now the foregoing Equations being Solved, and the values
of if Cf d. M reflored, it will be found that y =s ^^^"i^ .

aaaoi » laaoi > ** ~~ aaaot •

From which we may conclude, that the fcveral Expeftations
of B, C, D, A are refpeftively, i^y?, 4 x -i^ + -ilill-^.

Wi/,,4^ ^ + -il22i_^. rA/ri^,4;!k.+ J^p,

The

■ I l yj w p wy^riB^f w. i . i ^ 1 m i ii ^ i y j >ii .^iji. yi 1 nn i ^m pn^j . ■ n'pnn ■ ■ i w ■ n. i !■■ i n

■tfhiAa^W^ ■<

The Doctrine 0/ Changes* $$

The Expe£tations of the Gamefters being thus found, ic
will be neceflary to find the Risks of their being Fined, or
othcrwife what fum each of them ought juftly to- give to
have their Fines Infured. In order to which, let us lorm fo
many S'>^hemes tor cxpreding the Probabilities of the FiQe»
as are fufBcicnt to find the Law of their Procefs.

And Firfly we may obfcrve, that upon tiie fuppodtioa of
B beating A the fird Game, in confcquence of which A is •
to be Fined, B and C have one Chance each for being Fined
the fecond Game, as it thus appears

I

BA
LB

BA

BC

Sccondtj, that C has one Chance in four for being Fined
the third Game, B one Chance likewife, and D two; ac-
cording to the following Scheme,

3
3

BA BA BA

CB
DC

CB
CD

BC
DB

BA

BC
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 following Scheme,.

I

3

4

BA BA BA BA BA BA

CB
DC
AD

CB
DC
DA

CB
CD
AC

CB
CD
CA

BC
DB
AD

BC
DB
DA

N. B. The two Combinations BA, BC^ BD, AB, and BAt
BC, BD, BA are omitted in this Scheme, as being fuperflu-^
ous ; their difpofition fliewing that the Set mud have been
ended in three Games, and confequently not a^cding 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 circum-?
ftaoce is but -^.

Thefe

■ !i .i i i mm j ^ mmm <9' ^ i

■ "WI W

•^ !m > I I I I m i9—mm

!■ i . i ^ iwi'ifwiwwwia jw. < H| i .; I WI

i LUg ii UJ it l lWWI^

$6 Tlje Doctrine of Chances.

Tlicfc Schemes being continued, ic will cafily be. perceived
that the circumftanccs under which the Gamcfters find them*
fclvcs, in rerpeft of their Risks of being Fined, ftand i elated
to one another in the fame manner as were their Probabilities
of Winning ; from which confideration a Table of the Risks
may calily be compofed as follows.

A TABLE o//?/5/r5&c.

iWI^BiBB

/

2

it

9

tit

4

tm

5

V

6

r*

7

Iff

S

Iff/

9

\&c. &c.\

B

2 ^

T-f

16 '
255 f^

8

12ti

^56

f

D

4
2

256*

32 *
32 «

12» «
25tf «

Wherefore fuppofing B' + B" + B"' A'C. C^ + C" + C''
&c. D' + D" + D"' &c. ^' + ^'' + A'" &c. 10 reprefcnt
the fevcral Probabilities s and fuppofmg that the feveral fums
of thefe Probabilities arc refpeSively equal to j', jc, ;^ v, wc
{hall have the following Schemes and Equations

B' =

fl*

B" ^

ET

El" =

-f-^" +

■A'

r'' =

-h^" +

■A"

B" =

; >r"'+.

■A'"

r =

-\-A^-\--

AT

8fC.

Hence y = -L. -u «L

C' =

* ^'- '^'^' J ■' iiii|pwpi7pwytp»>w— ^^— p»i.iia m^in^^iim

i^Fi.

<■■ '■»■

■•HmK^

Doctrine 0/ Chances.

91

c

ss

c

c

ss

c

•

C"

ss :

1

B"

D'

C""

•M^ •

I

3

r

+ -r

D"

c

ss

\
3

g'"

vr

C'

ss

1 _

3

ET

+ -r

iT

&C.

^

cc« =

X

2

*

-jf^.

4

D'

2IS

1/

•

D"

=5

p"

/

D'"

=

I
3

cr

+ 4-

e

If"

=

3

r

+-T-

ET

P"

=

I

3

c""

+ -T

■ r

D"

=r

I
3

c

+ 4-

Ef"

&c

^

:nce v

'«

1
4

-+

4-*

+ -5

A'

».

^

p

A"

ss

A

If

r

:=

1 _

3

•D^

+ H

r-C

A"'

s

1

2

r

+ -i

rC^

A

=»

I

2

D- + J

h-cr

A'

S^S

s

D^

+ -i

-cr

8fc

ice X

=

t
2

• 4

z.

Cc

The

■ ■iiiiiM i»,r ■■

•••^^r<ww»'""W'

^p<wHMq^ni " ^

..,111 II Mfun — IT II irTpryjp^wj

^8 The Doctrine (/Chances.

The foregoing Equations being refolvedi we (hall have

y s .111, « = -in, V = -H* . X = J^i^

Let every one of thofe Fractions be now multiplied by /,
and the Produfts -li^n -^-L^-a -Hif^, lll.p will exprefs

the rcfpeftive Risks of U, C, jD, ^, or the lums they might
juftly give to have their Fines Infured.

But if from tlie feveral Expeflations of the Gamefters there
be fubtrafled, Firft^ the fums bv them depofited in the be*
ginning of the Play, and Stcondly^ the Risks of their Fine%
there will remain the clear Gain or Lx)fs of each. Wherefore

From the ExpeQations of B = 4x-f^ + *^^^* ^

Subtracting his own Stake ss i

32201
and alfo theTum of the Risks s ^»t

«♦? r>

h

There remains his clear Gain =s ^-ZJ^ 4. »i»»

_ »49 ^ 2320I

From the Expeaations of C =s 4 x JLi-, + -ilui-^
Subtracting his own Suke r= i ^

and alfo the Sum of his Riskl s

Therd remains bis clear Gain ss

From the ExpeSatioos of D ss 4 x — 12_ 4. -37<oo^

Subtraaing bis own Stake ss i

and alfo the ium of his Risks s

There remains hb clear Gaio = «—

From the Expcfhtions of ^ s 4 x -ii_ ^ \%%ki ^

Subtrading his own Suke ss "^ '"'* ^'

and alfo the Sum of his Risks ae

Laftlyt the Fine due to the Stock ^ ^
by the k>(sof the firft Game. 3.

There remains his dear Gain as — — 4f „ ^^ 1472^

'49 3220I

The

mvaipi m^.

iJJ I >■ W" l ! I . .. i J.J l ty^tWL Ii |IL, .» WM.MiU I.WMPM# ., . s iWia flH

The DocTniNEi)/ Chances. 9^

The foregoing Calculation being made upon the fuppofi-
tion of B beating A in the beginning of the Play, which (iip*
poficion neither afTeds C nor D, it follows that the fiim of
the Gains between B and A ought to be divided equally 1
and their feveral Gains will dzm as follows^

Gain of

>** l^f 4til04 ft

^ i\9 • at2oi /^

D s= — ■■»« ■■ + »"» »

Sum of the Gains

If -11 U2±^p which is the Gain of ^ or A. be

made s o ; then f will be found =s - ^ - ^^^ : Fixmi whidi *

it follows, that if each Man^s Stake be to the Fine in the
proportion of x/oo to 19)7, then A and B are in this ode .
neither winners nor lofers ; but C wins - ' ■ , which D lo-

Its.-

And in- the like manner may«be found what the proportion
between the Stake and the Fine ought to be, to make C or
D play with(3ut Advantage or Difad vantage ; and alfo what ;
th's proportion ousht to be, to make them play with any^-
Ad vantage or Di fadvantage . given.

CorolUn L A fpedator R might at firfl: in confide^
ration of the Sum ^-r^jf paid him in hand, undertake to
furnifh the four Gamefters with Stakes, and to pay aU their
Fines.

CoroBarj II. If the proportion of Skill between the Game-
fters be given, then their Gain or Lo(s may be determi^
ned by the methods ufed in this and the precedii^ Fro*
blem.

Corollary III. If there be never fo many Gamefters play*
ing on the conditions of this Problem, and the propor*
tion of Skill between them all be fuppofed equal, then the ;
Probabilities of winning, or of being Fined, may be deters*
mined as follows.

Let-

'''" ' ■•'•""w— • *m •■ *~<i-i^n^Baa«aw<r^pv>w.nww*w«v**w«nr*'«V>"M*VW>*>«^i««*V""'w«"w«*<<"^P"i'*M'*

loo The Doctrine 0/ Chances.

Let F, 'C\ D\ E\ Fy A' denote the Probabilities which
B,C, O, £, F^ A have of wiuning the Sec, or of being Fined,
inany number of Ganges; and let the Probabihties of winning
or being b'ined in any number of Games lefs bv one than the
preceding, be denoted by B'' iP 2J'^ E^ F'' jPi And fo on*

Then I fay that.

F = -7-F'+ -rD^' + 4-c^' + -irfi^

Corolbrj IV. If the Terms ^, jB, C, D, J5, F&c. of a Sc-
ries be continually dccreafing, and that the Relation which
each Term of the Scries has to the fame number of preceding
ones be coniUntly expred by the fame number of given
Fractions -».,-«, -ri -7 &c. For Example, if E be equal
to -1- D + J- C + ^- B, and /^ be alfo equal to J- £ + J-D
+ I. C, and fo on: Then I fay that all the Terms Ad

infinitum of fuch Series as this, may be eafily fummed up,
by following the (leps of the Analylis ufed in this Prob-
lem \ of which feveral Inflances will be given in the Prob-
lem relating to the duration of Play.

And if the Terms of fuch Series be multiplyed refpedl*
ively by any Series of Terms, wbofe laft differences are equal,
then the Scries refulting fiom this multiplication is exaQly
fummable.

And if there be two fuch Scries or more, and the Terms
of one be refpeftively multiplyed by rhecorrefponding Terms
or the other, then the Series refulting from this multiplica-
tion will be exaftly fummable.

Laftly, If there be feveral Series fo related to one another,
that each Term in the one may have to a certain number
of terms in the other certain given porportions, and that the
order of thefe proportions bcconftant and uniform, then will
all thofc Series be exaQly fummable. The

MiWi

** ' * ^''^^■9*V>*'*^*^**nwfVf*^*^vw

• ••«••• • •••

• ••••!•• • • .'

• •••• •• •"

•*• , , •••• • ••, •» • • "

TAe Doctrine 0/ CHANCE-^y- ^.:i'»in'.:'-^

The foregoing Problem having been formerly Solved by
me, and Printed in the PhitcfofhicdTrdnfaSioHS N* 341. Dr.
Brook Tajtofj that Excellent Mathematician, Secretary to the
Rcy/U Society^ and my Worthy Friend, loon after communi*
cated to me a very Ingenious Method of his, for finding the
Relations which tlie Probabilities of winning bear to one-
anothcr, in the cafe of an cnuality of Skill between the Game*
iters. The Method is as follows.

Let BJCD reprefent the four Gamefters; let alfothc two
Rvd Letters reprefent that B beats A the firft Game, and
the other two the order of Play.

This being fup^ofed, the circumllanccs of the Gamefters
will be repretented in the next Game by BCD A or CBDA.

Again, the two preceding Combinations will each of them
produce two more Combinations for the Game following,
to that the Combinations for that Game will be four io all,
'Viz. BDAC'i DBACj and CDABj DCAB; which may be fitly
reprefented by the following Scheme.

BACD

BCDA

CBDA

^BDAC

DBAd

CDAB

DCAi

It appears from this Scheme, that if the Combination CBDA
Happens, which mull be in the fecond Game, then B will be
in the fame Circumftance wherein A was the Game before;
the conformity of which Circumftances lies in this, that B is
beat by one who was juft: come into Play when he engaged
him. It appears likewife, that if the Combination DBAC
Happens, which muft be in the third Game, then B is again
in the fame Circumltance wherein A was two Games be-
fore.

'But the Probability of the firft Circumftance is -1-, and

the Probability of the fecond is -i-.

Wherefore the Probability which B has of winning die

^Set, in any number of Games taken from the beginning, is -|-

of the Probability which A has of winning it in the fame

number of Games wanting one, taken from the beginning ; as

Dd alfo

mtv

w

wmm

m I II J » '

<p»n»fr

IWIVfWVIPMHWMMW"**

• • • •

. • •

• •• • • • •• • ••

:•^::^^ib4i;..;. -Zoe DoCTRINE o/ ChaNCES.

alfo -1- of the Probability which he has of winning ia the

fame number of Games wanting two. From which it foU
lows, that if the Probability which B has of winning the Set,
in Five Games for inftance, and the Probabilities which A
has of winning it in Four and Three, be refpcftively denoted
by B% J!'"^ A'\ we (hall have the Equation, B^ - \ A'"

+ -L- A'\ which is conformable to what we had found

before. And from the Infpeflion of the fame Scheme may
likewife be deduced the Relations of the Probabilities of win-
ning, as they lye between the other Gamefters. And other
Schemes of this nature for any number of Gamefters may
eafily be made in imitation of this, by which the Probable
litics of winning or being Fined may be determined by bare
Infpeftion*

PROBLEM XXXIIt

TWO Cdmeflers A dnd B, rphofe froPcriion ff Skii is ds
z tohj each having d cert din number of Pieces^ flgj ta^
geiher on condition thdt ds off en ds A Wins d Gdme^ B fbdi
give him one Piece^ dndtbdi ds often ds B IVins d Cdme^ A JbdS
give him one Pieces dnd thdi tkcf cedfe not to P/djf till fucb timo
ds either one or the other hds got sU the Pieces of his ddverfgrj.
Notp let us fuffofe two Speffdtors R dnd S to Idf d Wmr j-
bout the Ending cf the FUjj the prfi of them tdjing thai the Ply
wiO be Ended in d cert din number rf Gdmes which- he dff^ns;
the other Idfing to the contrdry. Whdt is the Probdbilitf thdi S
bds of Winning his Wi^er f

S GLUT I ON.

CASE I..

LET Two be the number of Pieces which each Game-
(ler has, let alfo Two be the number of Games about
; which the Wager is laid : Now becaufe two is the number

of Games contended for, let 4 + ^ be raifed to its Square^
viz. 44 -V X4^ + bb -, and it is plain that the Term %dh favours
Sf and that the other two are againft him^ and confequently

that the Probability he has of Winning is -^j^^^

CAS Ell

»m > w .j ii . pi I I . , , ,^ ^ ,

The Doctrine 0/ Chances* toj

c AS B II.

LET Two be the number of Pieces of each Gamefter,*
but let Three be the number of Games upon wbkh
the Wager is laid: Then A-^-b being raifed to its Cubt
viz. a} 4- jdat + ^abb 4- ^', it is plain that the two Terms
4' and b^ are contrary to Sy Hnce they denote the number of
Chances for winning three times together ; 'tis plain alfo that
the other Terms 344^^ 34^^ are partly for him, partly againft
him. Let thefe Terms therefore be divided into their proper
parts, viz. -xddb into ddb^ sbsj bddj and ^My into My bsAy Um-.
Now out or thefe Six parts there are four which are favou-
rable to Sy viz. abdy bddy dbby bdb or X44^+ xM ; from whence
it follows that the Probability which S has of winning his

Wager will be ^2JL!Lt^llL: Or dividing both Numerator

and Denominator by dJ^b, it will be found to be ^=^
which is .the (ame as before.

C A S^E in.

LET 'Two be the number of Keces of each Gameftert
and Four the number of Games upon which the Wager
is- laid: Let therefore 4 + ^ be raifed to the fourth Power,
which is 4« + 44'^ + 6ddbb + 44^' + K The Termft
4« ^- 4 4'^ 4. 44^' -V ^ are wholly againft Sy and the only
Term 6ddbb is partly for him, partly againft him: Let this
Term therefore be divided into its Parts, viz. ddbb^ dkdi^
abbdy bddby bdbdy bbdd ; and Four of thefe Parts 4^4^, dbbdy bddt^
bdbdy or 444^^ win be found to favours S; from which ir

follows, that his Probability of winning will be zij^^

C A SE IV;

I' F Two be the number of Pcices of ^cb Gamefter^ and
Five the number of Games about which the Wager
is laid ; the Probability which £ has of Winning his Wager
will be found to be the lame as . in the preceding Ca(e^'

^** 'TI^- GSNE^

4+#« •

Mil .iii^in.r,

f 04 T7}e Doctrine of Chancei

G ENERALLT.

Let Two be the number of Pieces of each Gamcfter, and
2 4- ^ the number of Games about which R and 5 contend,
and it will be found that the Probability which S has of

Winning will be I_-^^ t,^ • But if if be an odd Number,
fubflitute d — t in the room of it;

CASE V.

LET Three be the number of Pieces of each Gamefter,
and i + d the number of Games upon which the Wa-
ger is laid ; and the Probability which S has of Winning

will be im .J . But if i< be an Odd Number, you arc

to fubflitute ^—1 in the room of it.

C ASE \h

IF the number of Pieces of each Gamefter be more than
Three, the Expe£lation of S^ or the Probability there is
that the Play will not be Ended in a given number of Games^
may be determined in the following manner*

A Generd RV LE for Determiniffg whMt Froksbilitj there
is that the . Plof tPiU not be EttJed im d given number of
Gdmes*

LET n be the number of Pieces of each Gamefter ; let
al(b n+d be the number of Games given. Raife d+b
to the Power n^ then cut o(F the two cxtrcam Terms, and
multiply the remainder by dd -{- i.Jb 4- lib : then cut off again
the two Extreams, and multiply again the remainder by
^u + tdh \ hb^ ftill reje£ling the two Extreams, and fo on,
making as many Multiplycations as there are Units in -~^;

Let the lad Produfl be the Numerator of a Fra£lion whofc

Denominator is aiS^^'^^t and that PraSion will exprefsthe

Proba*

l«fp1MKaw*^VM*i|M^«mfn

The Doctrine o/Chances. 105

Probability required, or the Expeflation of S. Still obferving
that if ^ be an Odd Number, you write ir^\ in the room
of ir.

EXAMPLE L

LET Four be the number of Pieces of each Gamefter, and
Ten the number of Games given : In this Cafe ;i> = 4,
and » + ^ = 10. Wherefore ^ =s 6, and -|- ^ =s 3. Let

therefore 4+i be raifed to the Fourth Power, and rejeAing
continually the Extreams, let three Multiplications be made
by AA + xdh \ hh. thus,

ii4) + ^\h + daahh + t^h^ (+ *♦
Aa 4" 2a3 + bh

ifi^h) + 6a^hh + ^^^

aa + 2flfr + W
I4ii*3i)+ 2oa'3> + 1411434

48jyt8 + 68fl^*4 + 48as^s
a« + 'O.Ah + W

48^735) + 68a* fr4 + 4&f53f

496tf*t4+ i36tf5AJ + 96^43*

+ 48tf J4* + 68tf4t^ (+48tf*7

164^*44 + 2;2a'3' + 164^4^

Wherefore the Probability that the Play will not be ended
in Ten Games will be '^-^^'^'-^^li^^fj^'^^^^' , which ex-

predion will be reduced to -^4^ or -21-, if there be an c-

quality of Skill between the Gamefters. Now this FraAion
being fubtraflcd from Unity, the remainder will be -IS-

wliich will expre(s the Probability of the Flay Ending in

E e Ten

I J B« »P — .

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■y,..iill n I W fi »■ ii n w

i ■ ■ [ 1 . 1 111 j mmmmmf^

io6 The Doctrine o/Chances.

Ten Games: And confequently it is 3 f to 29, that two e-
c]ual Gamtfters playing together, there will not be Four Stakes
loil on either fide in Ten Games.

N. B, The foregoing Operation may be very much con-
tra^cd by omitting the Letters 4 and /, and reilorins them
after the lall Multiplication ; which may be done in this man-
ner. Make n + ^i — i = ^» and -^i + i = ^: Then an-

nex to the rcfpeftive Terms refulting from the laft Multipli-
cation the literal Produds j^^f, 4>-«*fti, 4>-**f*« &C
Thus in the foregoing Example, inllead of the firft Multi-
plicand 44'^ -^ Saahb + 44^S we might have taken only
4 + 6 + 4, and inftead of Multiplying Three times by
OA-y X 4^ + ^ we might have Multiplyed only by 1 4. x 4. r^
which would have made the lad Terms to have oeen 164 +
a}x -t- 164. Now fmce that n\^ ^ ^ and d ^ 6\ f will
be s 6, and f =s 4 ; and confequently the literal Products
to be annext to the Terms 164 + %\x + 164 will be re*
fpe£^ively 4^^, 4'^^ 4*^% which will make the Terms refult-.
ing from the laft Multiplication to be 1644^^ -)- X3X4'^
4- 1644^^% as they had been found before.

EXAMPLE m

LET Five be the number of Pieces of each Gamefter, and
Ten the number of Games given. Let alfo the propor*
tion of Skill between A and B be as Two to One.

Since jy is s f , and ;» + 1< rs 10, it follows that di& ^ ^^
Now d being an odd number muft be-leflened by Unity^^
and fuppofed s 4, fo that -^d =s x. Let therefore 44.^ be

raifed to the fifth Power ; and always rejeQiog the extreams^

Multiply twice by ad + x^ + ttj or rather by i -)- x •)- 1 ;

thus,

i) + S + io+io + s(+i 20+ 9j +3S + 20

I + 2+ I I + 2 + 1

y)+io+ 10+ J 20)+3j + 35 + 2O

+ 10 + 20+20+10 40+70 +70-K40.

+ 5 + 10 + io(+ J. ?o+jj+3s(+20

20 + 35 + J5 + 20 7S +125+125+75

Nov

l,*,V , , , „ ^ !■! , ,

m^

The Doctrine 0/ Chances. 107

Now. to fupply the literal Prbdu^^s that are waotiog, let
n + -i-J— 1 be made = />, and -i- J + i =s j, then/ will
be =s 6 and jT = j. Wherefore the Produfts to be annext.
wz.ifb9^ d^-'bt-^' &c. will become 4*i», 4»i% 4^i», 41**1
and confequendy the Expefhtioa of S will be found to be

■iTi>*

N. J?. When /i is an odd number, as it is in this Cafe,
the Expeftation of 5 will always be divifibic by ir+*. Where-
fore dividing both Numerator and Denominator by a-^^t^ the
foregoing Exprefllon will be reduced to

-iul^^*±2.s^^, or t54»^»x i"^_2^JT^

Let now 4 be interpreted by % and b by i^ and the Ex*
pedatioo of S will become

loo

PROBLEM XXXIV/

T//E fame Things being pivenas in the PreceMng ProUemi
to find the Exfelfation o? R^ or otherwije wbdt tie Probdhi*
liij is that the PUjmUheEnoiiin a givem number if Garner^

SO LU T I ON.

Flrfi^ It is plain that if the ExpeQation of S^ obtained bf
the preceding Problem, be fubcra£led from Unity, there
will remain the Expeflation of /I.'

Seeondlj^ Since the Expeflation of S decreafct continually
as the number of Games increafes, and that the Terms we
rejeded in the former Problem being divided by ma + xdb 4. bb
are the Decrement of his Expectation ; it f ollow s, that if thofe
rejeftcd Terms be divided continually by a-^b^ they will be
the Increment of the Expectation of R. Wherefore the Ex-
pectation of R may be exprcfied by means of thofe reje&ed
Terms. Thus, in the fecond Example of the preceding Pro-
blem, the Expe^arion of R expreded by means of the it«
jeCted Terms will be found to be

^ " ' " "*" ' ■ ' I ■■■■ I m il .mi ii fi iip m ^iii m ii w.in iii n III .^f p. .

loS 77)e Doctrine </ Chances.

In the like manner, if Six were the number of the Pieces
of each Gamcftcr, and the number of Games were Fourteen ;
ic would be found chat the Expeftation of R would be

And if Seven were the number of tlic Pieces of each Game*
flcr, and the number of Games given were Fifteen; then
the Expeftation of R would be found to be

N. B. The number of Terras of thefe Series will always
be equal to -i- d + i, if i be an even number, or to -lil*

if it be odd.

Thirdly^ All the Terms of thefe Series have to one another
certain Relations ; which being once difcovered, each Term
of any Series refulting from any Cafe of this Problem, may
be eafily generated from the preceding ones.

Thus in the firft of the two laft foregoing Series, the Nu-
roerical Coefficient belonging to the Numerator of each Term»
may be derived from the preceding ones, in the following
manner. Let Kj^ L^ M h^ the Three laft Coefficients, and
let N be the Coefficient of the next Term wanted ; then ic
will be found that N in that Series will conftantly be equal
ro6A/— 9L4.1/G Wherefore if the Term which would

follow ^/?^ » in the Cafe of Sixteen Games given,

were defired ; then make M zsl 4x9, X = 110, I^ = 17,
and the following Coefficient will be found 1638. From

whence it appears that the Term it felf would be -il2^^.

Likewife, in the fecond of the two foregoing Series, if the
Law by which each Term is related to the preceding ones

were

^ pyyipr>gWgyiW»i»1WWW*ftfT^y'' ■■■ ■. ■ ■'■WI W- ■ ^ ■ ■I P H H H * l ■ ii ■■ n w n i mja ii j ii n

• «

Tbe Doctrine qf Chances. io^

were demanded, it might be thus found. Let J^t A ^» *>«
the Coefficients of the three laft Terms, and N the Coefficient
of the Term defircd ; then N will in th at Series, con ftantly
be equal to 7M — 14 L+ 7 Kaor to M — tL+lCx 7.
Now this Coefficient being obtained, the Term to which it
belongs is formed immediately.

But if the general Law, by which each Coefficient is ge*
nerated from the preceding ones, be demanded, it will be
expreft as follows. Let jy oe the number of Pieces of each
Gamefter : Then each Coefficient contains

n times the laft,
•— It %~^ times the laft but one^

+ n }c -^ X -^^^ times the laft but two,

_ ;j X ^=^ X ^x-i^ times the laft tilt threci

jL n X Ji^ x-^n. x-^:::^ x*^:^ timesthelaft butfboi^

&C.

Thus the number of Pieces of each Gamefter being Sfac^
the firft Term » would be =s 6, the fecond Term m x «^iiX

would be s 9, the third Term » x -2^^=^ x -^ would be rs x;

the reft of the Terms vanifliing in this Cafe. Wherefore if
Ks L^ M are the three laft Coefficients, the Coefficient of the
following Term will bt6M — 9JL -t-x/0

Fourthljy The Coefficient of any Term of thefe Series may
be found, independently from any relation they may h4ve
to the preceding ones : In order to which it is to be d>ftr-
ved that each Term of thefe Series is proportional to thd^
Probability of the Plays Ending in a certain number of Games:
precifcly 1 Thus in the Series which expreffes the Expe£tatioa
of A, when each Gamefter is fuppofed to have Six Pieces^

K

the laft Term, being multiplied by the common MultipU
cator y^^ r fet down before the Series, that Is' the Pro*^

dua -liP^^jgSE., denotes the Probability of the Playi

F f End-

■^i<^"H>^ \ . w^"— ■« ■ I ■<wr<i^|wg»wMw«r«>..M>»^«»i«p»^iiiwiyii«»i^i^tnww*B*^wn.|

no ^be Doctrine 0/ Chancks.

Ending in Fourteen Games precifely. Wherefore if that Term.
were defircd which exprcflcs the Probability of the Plays End-
ing in Twenty Games precifely, or in any number of Games-
denoted by ;»+</, I fay that the Cocfiicient of that Term will
be,

-l^x-^x- *'*'^''' X "•^'^'^'^ X >'-*'^"""y &C. continued

to fo many Terms as there are Units in J^i +• i ^

L,x-lfi-x '"^^^' X ''+^^^ x "-♦'^"■^ &c.conti^

X X 2 3 4

nued to fo many Terms as there are Units in ^-^i + x — n.
^. -L. X -li. X »+^-' X '"^^^' X ""^^--^ &c.continu-

• X X 2 3 4

cd to fo many Terms as there arc Units in -i-i/ + i — x /r.
-~i-x-^x ''•^^•" X Ji±f^x.!Ll:^^ll-&c. con.

X X 3 3 4

tinued to fo many Terms as there arc Units in -l-^ + i — 3 «$

Let now ji + i be fuppofed =s xo, n being already fiip-
pofed =s 6, then the Coefficient demanded will be found fronr
the general Rule to be,

-7- X-T-X-s-^C-j-X-j-X-^X-j-X-y- — XJXJO

Wherefore the Coefficient demanded will be 23x5*6 — 18
s ^3258 ^ And then the Term it felf to which this Co.
efficient does belong, will be -^ml^f^iL^ Confequently the

Probability of the Plays Ending in Twenty Games predfcly
wiUbc J^x^iiS?;^

Fiftblj^ By the help of the two Methods explained In this-
Problem ( whereof the firft is for finding the relation which
any Term of the Series refulting from the Problem, has to a
certain number of preceding ones ; and the fecond for find*
ing any Term of the Series independently from any other
Term) together with the Method of fununiog up any given

ounH

. 1 . 1 j i . I ■ I J. ii ■ 1 1 1 " ' ' ' ' '

The Doctrine 0/ Changes: Tii^

number of Terms of thefe Series, (which (hall be explained
in its place) ; the Probability of the Plays Ending in any p*
ven number of Games, will oe found much more readily thair
can be done by either of the two firft Methods takea
fingly.

PROBLEM XXXV.

SVppofwg A dni B to flxf together^ till -fticb time as Four
Stakes Are Won or Lojl on either fide : What mufi te their pro* •
portion of SkiO, to make it as Probable that the Plsjf mil hi BnJ-^
ei in Four Games^ as not f

SOLUTION.

'T' H E Probability of the Play Ending in Four Games, if

^ by the preceding Problem '^'t:^^ ; Now becaufe, by

Hjpothejis, it is to be an equal Chance whether the Play
Ends or Ends not in Four Games ; let this expreffion of tlie
Probability be made equal to -l.* And we (hall have this

Equation "^T^J^l = -p, which, making b^a:: x, ^ is re-
duced to -^At == -p> or *♦ — 4 *' •— 6 zjc — 4* + X = o.

Let I %zc be added on both fides the Equation, then will
;&♦ — ^zf '^ 6z,z — 4«+ I be = i%zz»^ and extrafting
the fquare Root on both fides, it will be reduced to this
Quadrat ick Equation ;c^— xj&+ i =*Vix, whofedoubfe
Root is jc = 5, X74 and — i — . Wherefore whether the Skill

01 A be to that of B as 5.174 to i, or as i to ; x74, there
will be an equality of Chance for the Play to be Ended or
not Ended in Four Games.

PROBLEM XXXVf •

SVppo/ing that A and B Plaj tiB fucb time as Four States
are Won or Loft : What mufi be their proportion of SkiB^
to make it a Wager of Three to One^ that tie Plajf will ie End^
ed in Four Games f

S O L U-

.» •

<<*w<

' ■ ■ I I I ill ■ . II . iiii .njm yiii i ff I t\ t m att i m ifm^mnmmmmmmm$mmmmmm>9fmmmm

gi;i 7U Doctrine 0/ Chanccs.

SOLUT ION*

TH E Probability of the Plays Ending in Four Games,
ariOng from the number of Games Four, from tho
number ot Stakes Four, and from the proportion of SkxW

is ^^"1"^^ . The fame Probability arifing from the odds of
Three to One, is -^^ Wherefore -i^A^ = -i- and fup*
pofing ti di: liZy the foregoing Equation will be changed
into -i^jip = -1-, or «♦ — izzf-^iBzz — ix;e-l-.is3a

Let 56 zz be added on both fides the Equation, then we
(hall have jc* — ix*' + iSjt*— i»»+ 1 = $6zz. And
Extracting the fquare Root on both fides, we fliall have
;&s — 6 £ + I =f z^ $69 the Roots of which Equation will
be found 13407 and .-j-i—-. Wherefore, whether the Skill

of A be to that of £ as 13.407 to i, or as i to i3407,\is
a Wager of Three to One, that the Play will be ^ ended ia*
Four Games.

PROBLEM. XXXVH.^

SVffopnz that A dni B Pldf tiUfucb time as Four Siskes
dTi Won or Lojt \ Wh§t mufi h their Proportion tf Skit^
to make it an ejudl Wager that the Flaj mil te Ended in Six
Games f

S O L U T I O Ni

TH E Probability of the Plays Ending, in Six Games, arifing
from the given number Six, from the number of Stakes

Four, and from the proportion of Skill, is ^^^\ x 1 4. tj^,^

The fame Probability arifing from an ecjuality of Chance for
Ending or not Ending in Six Games, is equal to -^ from

whence rcfults the Equation -^jrr ^ » + "^ffl^ * v whidi

by making byaii i, « may be changed into the followingi
viz. *!:t-6*»— 13** — xo*»— i3«;+6«+ i = a

^^lm9f^tf^•mml91^fmmn^mmm^F9m''mmm'^9^|»mt<^^^^^l'^mmml^|^l^''9f^m^^'mmm^ml^mrt^|'^»*^ • ■■» ■li-ih ■'■«<■ '"

The Doctrine c/Chances. 113

la this Equattoo, the Coefficients of the Terms cqoalty
diftant from the E^ctreams being the (ame, let it be fumolcd
that the Equation is generated from the Multiplicatioa n two
other Equations of the fame nature, viz. ««— -jr«-(- 1 s o^
and ;&♦ + /*' + J** + /-c -t- I =s a Now the EquadoQ
refulting from the Multiplication of thefe two will be

*!

which being compared with the firft Equation, we (hall have

From hence will be deduced a new Equation, vuu j^ 4- 6 jpr
— 16 jr — 3x = o, one of whofe Roots wUl be %.^6^^i
which being fubftituted in the Equation j&c— «/« -V- i s 09
we (ball at laft come to the Equation zz — x.96i|4 jc 4- 1 s o,
of which the two Roots will be %.<76 and — i—^ It foDows

therefore, that if the Skill of either Gamefter be to that of
the other as x.f 76 to i ; there will be an equal Chance for
Four Stakes to be Loft, or not to be Loft, in Six Games. .

CoroltdTj. If the Coefficients of the l^tream Terms of an
Equation, and likewife the Coefficients of the other Terms
equally diftant from the Extreams, be the (amet that Eaua*
tion will be reducible to another, in which the Dimenuons
of the higheft Term will not exceed half the Dimenfions of
the higheft Term in the former.

PROBLEM XXXVIII.

SVppojiffg A affd B, ivbofe f report ion of Skill is ds z toh^ t$
PUj together tiU fuch time ds A either iVi/n d certain nam^
ber q of Stdkes^ or B fome other number p cf them : Whdi is
the Probdbilitj that the PUj mH not be Ended in d given nnm*
ber (f Cdmes f

SOLUTION.

TAKE the Binomial 4+^, and re)e£ling continually thole
Terms in which the Dimenfions of the quantity d ex-
ceed the Dimenfions of the quantity b by f, rgeQing alfo

G g thoTe

<Wi

tr^^^tm'^mrmt

•vvn*«.HnF«^Nn«a«(«*MWPm^pqp

^fi

114 ^^^ Doctrine 0/ Chances.

thofe Terms in which the Dlmendons of the quantity b ex-
ceed the DimenHons of the quantity d hy pi multiply con«
fiantly the remainder by 4+^, and make as many Multipli-
cations, as there are Units in the given number of Games
wanting one. Then fhall the laft Produft be the Numei-ator
of a Fraftion exprefling tlie Probability required ; the Deno^
minator of which Fraaion always being the Binomial 4 + *
raifcd to that Power which is denoted by the given num*
ber of Games.

EXAM? L E.

LET ^ be = 5, f = 1, and let the given number of
Games be s 7. Let the following Operation be made
according to the foregoing direftions.

5 aihb + 8 otAi •\^ (lab*
a-\-b

S<**bb) +13 aibi+ 8 aab*

i^a^bi + 21 tfJ3*f + 8 aabs.

• k

From this Operation we may conclude, that the Proba-
bility that the Play will not oe Ended in Seven Games

is equal to ^LifS!£±^i.f!i!. ^ Now if an Equality of SkiH

be. fuppoled between A and B^ the Expreilioa of this Proba-
bility

, - , ..^,.., . , ^ .1 III . jiK m I I iiii . I - III II I ' II I — — — ■ " ■ ' ■■ " ■'■ «" ■■' "^n*

The Doctrine i/ Chance^:- 115

bility will be reduced to ' ? + y or -ij- : Wherefore the

Probability that the Play will End in Seven Games will
be -4^ : from which it follows that' *tis 47 to 17 that in Sc>

ven GameS| either A wins two Stakes or B wins three.

PROBLEM XXXIX.

THE fame Thingsbein^fupPos^iAS in the Preceding PrMem^
to fitti the Probabilitj of the PUjs being Ended in d gi^
ven number of Games.

SOLUTION.

»

F/r7?, If the Probability of the Plays not being Ended in •
the given number oi Games be fubtra£ted from Unity,
there will rema'm the Probability of its being Ended in the
fame number of Games.

Secondly^ This Probability may be exprefled by ipeans of
the Terms rejedled in the Operation belonging to the prece-
ding Problem ; Thus, if the number of Stakes be Three and
Two, the Probability of the Plays being Ended in S^vea .
Games may be ezpreifed as follows.

Suppofing d and b both equal to Unity, the fum of the
firft Series will be ss -|^, and the fum of the fecond will

be =s ol.; which two fums being added together, the ag-
gregate -^ exprefTes the Probability that in Seven Games

cither J fliall win Two Stakes or B Three.

Thirdlj^ The Probability of the Plays being Ended in a
certain number of Games, or fooner, is always compofed of a
double Series, when the Stakes are unequal ; which double
Series is reduced to a fmgle one, in the Cafe of equality
of Stakes.

The firft Series alwaj^s exprefTes the Probability there is that
^, in a given number of Games, or fooner, may win of B the

num^

■ ^" ^1 w n

'■I I I ! i . | p M*.<>ypf>yiiiii iw I - ' t i %wy!*»qpBrfW|qpy^WlWW»Wppj|

ti6 The Doctrine of Chances.

number q of Stakes, excluding the Probability there is, that
S, before that time, may be in a circumftance of winning
the number f of Stakes. Both which Probabilities are not
inconOftent together ; for ^ in Fifteen Games, for Inftance,
or fooner, may win Two Stakes of B, tiiough B before that
time may have been in a circumAance of winning Three
Stakes of A.

The fecond Series always expreffes the Probability there is
that By in that given number of Games or fooner, may win
of-^ a certain number p of Stakes, excluding the Probability
there is that ^, before that time, may win of B the number
q of Stakes.

The firft Terms of each Series may be reprefented refpeft-
ively by the following Terms.

Each of thefe Series continuing in that regularity, tHl fuch
time as there'be a number p of Terms taken in the firft, and
a number jq of Terms taken in the fecond ; after which the
Law of the continuation breaks o£

Now in order to find any of the Terms following in either of
thefe Series, proceed thus i\ttf + q — x be called /; let the
Coefficient of the Term delired be T; let alfo the Coefficients
of the pr^xding Terms taken in an inverted order be <S, /T,
4^ P&c. Then will T be equal to /5 — irLx-I^A + ^

X izixiiiL/2.-^x-iiix^xIiiP&aThusif/be=3,

and f s ly then / will be3-{-i — 2=33. Wherefore /5*—
LzJLx-!^^x ^ would in this Cafe be equal to 3 iS — /t;

which fbews that the Coefficient of any Term deOred would
be conftantly three times the laft, mhus once the la ft but one.
To apply this, let it be required to find what Probability
there is, that in Fifteen Games or fooner, either J fhall wia
two Stakes of J?, or B three Stakes of ^; or which is all on^
to find what Probability there is, that the Play fhall end in
Fifteen Games or fooner, A and B refolving to Play, till fuch
time as A either wins three StakeS| or B twa

Let

B**'*M

I'll i . ^ mmp i iPH w ■■■■■'^■>*"^^ip^^— — w»i w I I .1 ^ M iii j. ,..»i j ,»,

The Doctrine 0/ Chances. 117

Let Tuo and Three, ia the two foregoing Scries, be Tub-
flitutcd refpeftivcly in the room of q and^; then the three
firfl: Terms of the firft Series will be, fetting afide the com-

mon MultipHcator, i + -'jm + "737^* Likewifc the two

(irft Terms of the fecond will be i + "^TT?* ^^^ becaufe

the CocfHcient of any Term deQred in each Series,- is re-
fpeftively three times the lad, minus once the lad but one,
it follows, that the next Coefficient in the firft Series will
be I), and by the fame rule the next to it 34, and fo on.
In the fame manner the next Coefficient in the fecond Se-
ries will be found to be 8, and the next to it xi, and lb on.
Wherefore, reftoring the common Mukiplicators, the two Se-
ries will be

If we fuppofe an equality of Skill between A and B, the
fum of the firft Series will be -iiiL|-, the fum of the fecond

will be ^^^^\\ > and the aggregate of thefe two fums will

be 1^^^^ , which will exprefs the Probability of the Plays

Ending in Fifceen Games or fooner. This laft Fraflion be-
ing fubtrafted from Unity, there will remain -j^^t which
exprefTcs the Probability of the Plays continuing for Fifteen
Games at lead: Wherefore 'tis 31 171 to 1597, or 39 to x
nearly, that one of the two equal Gamefters, chat (ball be
pitcht upon, (hall in Fifteen Games, or fooner, 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 pre-

H h fixe

ffp*w™R«^lP^ir"ws^"^"*»^«^^i*^^WMP^'W*^''^

•••-**""'-i%|?«[

ii8 The Doctrine 0/ Chances.

fixe to it, being added together, muft either equal the num«
ber of Games given, or be lefs than it by Unity. Thus, in
the firft Scries, the Index ix of the Denominator of the laft
Term, and the Index a of the common Multiphcator being
added together, the fum is 14, which is lefs by Unity than
the number of Games given. So likcwife in the lecond Series, '
the Index ix of the Denom'mator of the laft Term, and the
Index ; of the common MukipHcator being added together,
the fum is 15, which precifely equals the number of Games
given.

It is carefully to be obferved, that thefe two Scries taken
together, exprefs the Expe£lation of one and the fame Per font
and not of two difierent Perfons ; that is properly the Ex-

gelation of a fpeflator who lays a Wager that the Play will
e Ended in a given number^ of Games. Yet in one Cafe
they may exprels the Expeflations of two different Perfons:
For Inftance, of the Gamefters themfelves, provided that both
Series be continued infinitely ; for in that Cafe, the firft Se-
ries infinitely continued will exprefs the Probability that the
Gamefter A may fooner win two Stakes of £, than that he
may lofe three to him : Likewife the fecond Series infinitely
continued will exprefs the Probability that the Gamefter B
may fooner win three Stakes of ^, than that he may lofe
two to him. And it will be found, when we come to treat
of the method of fumming up thefe Series, that the firft Series
infinitely continued will b e to the feco nd infin itely c ontinued,
in the proportion of mm x aa-^-ab-ybb to t^ x 7'^T ; that is,
in the Cafe of an equality of Skill, as three to two ; which
is conformable to what we have faid in our IXth. Problem..
Fourthly^ Any Term of thefe Series may be found indepen-
dently from any of the preceding ones: For if a Wager be laid
tiat A (hall either win a certain number of Stakes denomina*
ted by a, or that B (hall win a certain number of them deno-
mio I ted by /, and that the number of Games given be expre(fed
^> }-t^ i then I fay that the Coefficient of any Term in the
fii ft Series, anfweringto that number of Games, will be

+ -^x-^x y-^^^' X T"*-^^' X y-*-^'? &c conti-
filled to fo many Terms as there arc Voits in ^i .+ t.

■ J ■ ^— .^— ^ , . _ ^ ^^_ ^ _^ — ^_^^ [!..__ I . I -f«»w--

I

A

The Doctrine (/Chances. ii^

-- J-x Jd^ X i±fL=:i-x i±(=i2- X liiilL &c.cooti-

ill % 4

nued to fo many Terms as there are Units in .-i^i 4- i — ^,

ed to fo many Terms as there are Units in -J^.^/ ^ i .^^ -« ^1

i-Kl4^x^±4^xi±f=^x liiHL &C- contH

1113 4

nued to fo many Terms as there are Units in -i. ^ ^ i — . %f — ^ •

+ 4-X-S4i^x-ii^x.i±^ &C- continued

to (b many Terms as there are Units iax4-\- 1 — » »—■ % j,

ed to fo many Terms as there are Units in -L</ 4. i — 3^ — if.

+ -L^^ll±Jt. . T-t-^-. y-<-rf-« yfrf-} ^q. COntH

nued to fo many Terms as there are Units in ^i -}- 1 — 3/— Jf;

&C.

And the fame Law will hold for the other Series, calling
/+^ the number of Games given, and changing f into f^
and p into f , as alfo d into A

But when it Happens that d is an odd number, fubfti^
tute d-^i in the room of it, and the like for A

PROBLEM XL.

IP A d^d B, whofe frofortion of Skill is fuffefed ds Z toh^
pld)/ together : ff^hat is the ProtMitj thdt one ofthem^ fnf^
pofe A, msj in d number of Games not exceeding d number given^
win of B d certdin numberof Stakes f Ledving it tpholly indiffe^
rent whether B before the expiration of thofe Games may or mdf
not have been in a circumjldnce of winning the fame^ or anf othlet
number of Stakes of A.

SOLUTION*

SUppofing ;» to be the number of Stakes which ^ is to
_ wm uf B^ an J ^+^che given number of Games ; let a-^-b
be raifed (O tiie Powa wiiole Index is n';\^d : Then if d be.

I I IIIPBB^ n il _ __^.^_^

120 The Doctrine of Chances. «

an odd Number, take (b many Terms of that Power as there
are Units in -^i^; take alfo as many of the Terms next

following as have been taken already, but prefix to them, in
an inverted order, the Coefficients of the preceding Ternis.
But if 1/ be an even number, take fo many Terms of the faid
Power as there are Units in -i-J+ 1 5 then take as many of
the Terms next following as there are Units in JL.d^ and

prefix to them, in an inverted order, the Coefficients of the
preceding Terms, omitting the laft of them ; and thofe Terms
taken all together will compofe the Numerator of a Fraction
exprcffing the Probability required, its Denominator being

EXAMPLE I.

SUppofing the number of Stakes which A is to win to be
Three, and the given number of Games to be Ten ; let
4+^ be raifed to the tenth Power, vis^ ^^''-t 10 a^^-{- 4; a^lh

4. 45 dtb^ -V- i04^ + ^*^ Then by reafon that /f is =s }
and ;y-t- J s zo, it follows that ^ is s 7, and ^+' - s 4.

Wherefore let the Four firft Terms of the faid Power be
taken, viz. ji'* + 104*^ + 4$ 4'^^+ 1104^^% 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
Terms following with their new Coefficients, will be 1104^^
+ 45 4'*^ + i0 4*i* + 1 jj bf. And the Probability which
A has of winning Three Stakes of B in Ten GameS| or foo*
ner^ will be exprefled by the following Fradlioo,

which, in the Cafe of an equality of Skill between A and B^
will be reduced to -aii- op -ll..

1024 *'* ja

EX-

■ m ! ij I I" . '" ■ ■ ", ' ! ■ ■ ''

m v mw mm m y ^■iwiqib . i i mM iyyiP— ii—W^ 'mmt w mit li I ■^mw^'***"

Ti^^ DoCtRINBd/CHANCBS. 1*1

I

EXAMPLE U.

C UppoHng the number of Stakes which A is to win to be
^ Four, and the given number of Games to be Ten ; let
4+^ be raifed to the tenth Power, and by reafon that ;y is = 4,
and ft-^-d = lo, it follows, that d — 6 and -JUrf^- 1=4;

wherefore let the Four firft Terms of the faid Power be ta-
ken, viz. 4'* 4- I0 4* ^ -i- 4^ «• W ^ no a7 1^ ; take alfo
Three of the Terms following, but prefix to them, in an in-
verted order, the Coefficients of the Terms already taken,
omitting the lad of them. Hence the Three Terms following
with their new Coefficients will be 4^ «• ^ -f- 10 j* i^ -t-i 4♦*^
And the Probability which A has of winning Four Stakes of
By in Ten Games or fooner, will be expreffra by the follow-
ing Fraftion

g'*>+ io<i»5 + ^^a^bb+ notf^^i^- 4ya<M+ iOtf>^^ 4" I tf^^^

which, in the Cafe of an equality of Skill between A and Bf
will be reduced to -H^l^ or-rA-

S024 ^* las

Amb(r SOLUTION.

CUpponng,as before, that n be the number of Stakes which
^ ^ is to win, and that the given number of Games be j»+if
the Probability which A has of winning will be exprefled by
the following Series, viz^

■ i»X»-f-3j<4t^ j^ »x»4-4>

which Series ought to be continued to fo many Terms as
there are Units in JL^d + 1 ; always obferving to fubfli-

tute d — I in the room of ^, in cafe d be an odd number, or
which is the fame thing, taking fo many Terms as there arc
Units in -itJL.

I i Now

iR ^ i i im iip I w wiiii i> ii n )i H M »lf i ipili *. III . " ' H W^ J" ■ iJi y . fPffw^WPW

.'.tjr-

122 The Doctrine 0/ Chances.

Now fuppofing, as in the (irft Example of the preceding
Solution, that Three is the number of Stakes, and Ten the
given number of Games, as alfo that there is an equality of
Skill between A and £, the foregoing Series will become

as before.

I 4-

4

•-+-

9 _

16

1 ^M II

REMARK.

MOafieur de Mmmortj in the Second Edition of his
Book of Chances, having given a very handfom So*
lution of the Problem relating to the duration of Play, (which
Solucion is coincident with that of Monfieur Nicolds BernouU
Ij^ to be fcen in that Book ) and the Demonftration of it be-
ing very naturally deduced from our firil Solution of the fore-
gom^ Problem, I thought the Reader would be well pIeaC:d
to fee it transferred to this place.

Let it therefore be propofed to find the number of Chan-
ces there are, for A either to win Two Stakes of iS, or for B
to win Three of A in Fifteen Games.

The number of Chances required is exprefled by two Bran-
ches of Series ;. all the Series of the firft Bianch taken together
exprefs the number of Chances there are for A to win Twa
Stakes of B, exclufive of the number of Chances there are
for Bf before that time, to win Three Stakes of A. AH the
Series of the fecond Branch taken together exprefs the num-
ber of Chances there are for B to win Three Stakes of A^
exclufive of the number of Chances there are for A^ before
that time, to win Two Stakes of &

r

Firjt Branch of SER IBS.
iitJtfM^ aM^> a"^s a"^« a^^i^ a^h^ aHf a'Jb* a^b* a^b^^ a^b*^aib^*a*b*t

l+l5+l05+455+i3^5+50O3+5oo5+50O5+3O03+i365+4J5+i05+i5+l
— 1— 15- 105-455 — 455 — loj— 15— I

+ 1 + 15 + 1J+ I

Second Branch of S E R I E S.

i»5 *Mj 3«Jj» 6»»aJ i»»tf* *«•«» b^a^ b^tfl ^7a* b^A* b^a^^ 3*tf" 3itf««

^ I— ly-r 105-4JJ— 1365-455- loj— ij— I

.+. 1 + IS + I Th*

^yn.f ^pofwupipr— ^— wiw>^""> ^i" " -• ■ - ~' -t ■'^>— w ^ -..w n' .^fteam^^^nn^t^^iv^^t^nm^t^^itmtmfmtmimc ^ ' ^ ^-^i ■ ^ '^i<i .j w

Mil •

The Doctrine e/ Chances. laj

The literal Quantities, which are commonly annext to the
numerical ones, are here written on the top of them; which
is done, to the end that each Series being contained in one line,
the dependency they have upon one another, may thereby be
made the more conipicuous.

The (irft Series of the Hrft Branch exprefles the number of
Chances there are for A to win Two Stakes of S, including
the number of Chances there are for £, before the expira*
tion of the Fifteen Games, to be in a circumftance of wmning
Three Stakes of A *, which number of Chances may be de«
duced from our foregoing Problemr

The fecond Series of the firft Branch is a part of the firdj
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 firft Series, to win Two Stakes of B. It is to
be obferved about this Series, Firft ^ that the Chances of E
expreflcd by it are not rcftrained to Happen in any Order,
that is, either before or after A has won Two Staxes of B.
Secondly^ that the literal Pioiu£ts belonging to it are thefame
with thofe of the corrcfponding Terms of the firft Series.
Thirdly^ that it begins and ends at an interval from the firft
and laft Terms of the firft Series equal to the number of Stakes
which £ is to win. Fourthbj 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 nurn*
bers return in an inverted order to the end of that Series: .
Which is to be underftood in cafe the number of its Terms
fhould Happen to be even, for if it (hould Happen to be
odd, then thai order is to be continued to the greateft half,
after which the return is made by omitting the laft number.
Fifthly^ that all the numbers of it are Negative.

The Third Series of the firft Branch is a part of the fe-
cond, and expreftes the number of Chances there are for A
to win Two Stakes of By out of the number of Chances
there are in the fecond Series, for £, to win Three Stakes of
A\ with this difference, that it begins and ends at an inter-
val from the firft and laft 'Terms of the fecond Series, equal .
to the number of Stakes which A is to win i and that thd
Terms of it are all Fofitive.

It

^w^^i*wp^i*w»w(M»*»»irw»r'-»*"w^.^(9^w^w*'

II . . I II I " 1 1

124 ^^^ Doctrine 0/ Chances.

Ic is to be obferved in general that, let the number
of ihefe 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 i? is to win ; and
that the Interval between the beginning of the fecond and
the beginning of the third, is to be equal to the number
of Stakes which A is to win; and that thtfe Intervals re*
curr alternately in the fame Order. It is to be obferved like*
wife, that all thefe Series arc alternately Pofitive and Nega*
tive.

All the Obfervations made upon the fird Branch of Series
belonging alfo to the fecond, it would be ncedlefs to fay any
more of them.

Now the fum of all the Series of the fii ft Branch, being
added to the fum of all the Scries of the fecond, the aggre-
gate of thefe Aims will be the Numerator of a Fradlion ex-
prefltng the Probability of the Plays terminating in the given
number of Games; of which FraQion the Denominator is the
Binomial A-^h raifed to a Power, whofe Index is ecjual to
that given number of Games. Thus, fuppofing that, m the
Cafe of this Problem, both 4 and ^ are eaual to Unity, the
fum of the Series in the firft Branch will be 18778, theium
of the Series in the fecond will be 1x393; and the aggre-
gate of both 3 1 171: And the Fifteenth Power of x being
3x7^89 it follows, that the Probability of the Plays termina-
ting in Fifteen Games will be .-illl!., which being fub-

traOed from Unity, the remainder will be J J^^^ - : From

whence we may conclude, that ^is a Wager of 3*1 1 7 c to 1 597,
that either A in Fifteen Games ftiall win Two Stakes of B^
or B win Three Stakes of A : Which is conformable to what
we had before found in our XXXIX/ifr. Problem.

PROBLEM XLL

To Fini wbai ProhabUitj there is^ that in s given number
cfGimes^ A my be mnner ofd certain number q of Stakes ;
and at Come other time^ B msj likewife be winner of the number p
of Stakes^ fo thai both circumftances maj Hapfen.

SOLU.

■pill !■ *^ .\.^ m \ ^*%nm^mmmmii'w<mr9mmmtm^9mmfmmFt9mm^ n ■■ i - 1 1 . "" J " jl ■ ' ■■

The Doctrine ^Chances. 125

SOLUTION.

FIND, by our XUh Problem, the Probability which A
has of winning, without any Limitation, the number f of
Stakes : Find alfo by our XXXIV/^ Problem the Probability
which A has of winning that number of Stakes before B may
Happen to win the nuniDer p ; then from the firft Probability
fubcrafling the fecond, the remainder will exprefs the Probabi*
lity there is, that both A and B may be in a circumftance of
winning, but B before A. In the like manner, from the proba*
bility which B has of winning without any Limitation, fub*
trafling 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 circumftaoce of winning, but A before B.
Wherefore adding thefetwo remainders together, their fum
will exprefs the Probability required.

Thus, if it were required to find what Trobability there i^
that in Ten Games A may wia Two Stakes, and that at fome
other time B may win Three. The firft Series will be found
to be,

The fccond Series will likewife be found to be

The difference of thcfe Series being j^. x ^^^ j^

exprelTes the firft part of the Probability reguircd, which, ia
the Cafe of an equality of Skill between the Gamcftersi^ would
be reduced to —V

The Third Series is as follows.
The Fourth Scries is

ii+*'' • s-^-ki* ^ ^i+iT* •

The differcnceoftbefetwoSeries being '4-?x*^^ ' ^''*'

.- - ^"r»i 4*|*#i

• -f-L

K k expreP

■ II jui ;■» m m* t ■■ ) m'mtlft^mmfm^|fgm^lJlfm m\ l|^ m . i m ■■ nm m$ * *9F^mVmmnV^

126 The Doctrine 0/ Chances.

exprelTes the fecond part of Probability required, which, in
the Cafe of an cqirUity of Skill, would be reduced to -U-,

Wherefore the Probability required would in this Cafe
be ^ -4. -iJL. •= ~^. Whence it follows, that \is a Wa-
ger of 49 jr to 17, or of 29 to I verjr nearly, that in Ten Games,
A and B may not both be in a circumftance of winning, viz.
A the number y, and B the number f of Stakes. But if, by
the conditionsof the Problem, it were left indifferent whether
A or B fhould win the Two Stakes or the Three, then the
Probability required would be increafcd, and become as fol-
lows, vie

which, in the Cafe of an equality of Skill between the Game-
flerS| would be the double of what it was before.

PROBLEM XLII.

To Fini what Probabilitj there is^ that i^ d given ntmu
her of Gatms^ A mAj ipin the number q of Stakes ; mtb
this farther eondition^ that B, during that whole number of Games^
mn) never have been winner of the number p of Stakes.

SOLUTION.

FRom the Probability that A has to win without any limi^
tation the number q of Stakes, fubtrafl the Probability
there is that both A and B may be winners, viz^ A of the
number ^, and B of the number f of Stakes, and there will
remain the Probability required.

But, if the conditions of the Problem were extended to
this alternative, o/i^. that either A (hould win the number
f of Stakes, and B be excluded the winning of the number^;
or that B Ihould win the number f of StaKes, and ^ be ex-
cluded the winning of the number ^, the Probability that
either the one or the other of thefe tuo Cafes may Happen,
will eafily be deduced from what we have faid.

LEM.

TtT -■■■:■•*■.-

-• ■

— ■> ■ ■■.^■- ■-"

V

The Doctrine 0/ Chances. 127

LEMMA I.

I IT any Series of Terms^ VDhereof the prfl Differences are eqstat^
the Third Term tvill be twice the Second^ minus once the Firfi ;
and the Fourth Term IiLwife iviff be twice the Third^ minus once
the Second : Each foBoiving Term being always related in the fame
Manner to the two preceding ones. And as this relation is exfref
fed by the two tT umbers x — i, / therefore call thofe Numbers
the Index of that Relation.

In any Series of Terns Sy whofefecond Differences are cqual^ the Fourth
Term will be three times the Thirds minus three times the Second^
plus once the Firjl : And each Term in Jucha Series is alwys rr-
lated in the fame manner to the three next preceding ones^ ac^
cording to the Index 3 — H"'- Thusy if there be a Series of
Squaresyf'ch as 4, 1 6, ) 6, 64 , 1 00, whofefecond differences are known
to be equal when their Roots have equal Intervals^ as thy have
in this Caftj it will be found that the Fourth Term 64 // =
3x36 — 3 X 16 -h 1 X4, and that the Fifth Ttrm 100 // =
3 X 64 — 3x36-4-1x16. In like manner J if there snere a &•
ries of Triangular number s ^ fucb as 3, 10, xi, 36, J J, wbofeji"
cond Differences are known to be equals when their fides have equal
Intervals^ as they have in this Caje^ it iviObe found that the Fourth
Term /i = 3 x xi — 3 x 10 -4- i x }, and that the Fifth Term
is ■= 7 X ^6 — 3XXI-4-IXIO; and fo on.

' So liktwtfe^ if there were a Series of Terms whofe Third Diffe^
rences are equals or whofe Fourth Differences are -:= o\ fuch as is
a Series ofCubey or Fyramtdal numbers^ or any other Series of num*
bers generated bj the Quantities a A' -4- bxx -4- ex -h d, when a, b,
C, d being conflant Qusntities^ X is interpreted fuccefpvely by the
Terms of any Arithmetic Pro^refpon: Then it wtll be found that
anji Term of it is relaxed to the Four next preceding ones^ accor^
ding to the following Index^ viz. 4 — 6-4-4—1, whofe parts are
the Coefficients of the Binomial a — b raifed to the fourth Poiper^
the firjl Coefficient being omitted.

And generally^ if there le any Series of Terms whofe h(l Diffe^
rences are = o. Let the number denoting the rank of that dtffe^
fence be n; then the Index of the Relation of each Term to. as
many of the preceding ones as there are Vnits in n, will be ex*
preffed by the Coefficients of the Binomial a — b raifed to the
Power n, omitting tbefrfi. But

■< i . i ■»»■ > —IB > f ■ ^w^w^^w»p^^^■1^|p^^^q^■^H<w■>w^lT^wiWg*'f ' " ■ * ' I' '" ^ " ' '"• '^WP'ySffPWi^

f c8 The Doctrine i)/ ChanC'Es.

But if the ReUtionof dnj Term of a Series to s conjldnt nam*
ber of preceding Terms ^ he exfrejfed by atij other Indices thdn thofe
which are comprifed under the foregpiag general Law i or cven tff
thofe Ifidices remaining^ any of their Signs -{- or "^ be change J^
that Series of Ternts tviU have none of its differences equal to
tiothing.

LEMMA IL

IF in any Series^ the Terms A, B, C, D, E, F &e. be conti*
nually decreajingj and be fo related to one another that each of
them may have to the fame number of preceding^ Terms a certain
given Relation^ alwys expreffible by the fame Index; I fajy that
ahe fum of all the Terms of that ' Series ad infinitum msy always
be obtained*

Firft, Let the Relation of each Term to the two preceding ones
be expreffed in this manner^ viz. Let Q ^f = mBr— n Arr ;
and let D Ukewife ^r = m Cr — n Err, and fo on : Then will

the fum of that Infinite Series be equal to ^,1 ^Jal^rt *
.Thus J if it be fropofed to find the fum of the following Series^

A B C D E F G

viz. ir-4-3 rr-4- 5 r*-4-7r* + 9r^+ IX r*-|. xj r^ drc.

whofe Terms are related to one another in this manner^ viz. C =
xrB — irrA, D = xrC — 1 rv^&e. Let m andtk be made
refpclfively equal to x and x, and thefe Numerical Quantities be^
iM Subjiitutedj in the room of the literal ones^ in the general
Theorem^ the fum of the Terms of the foregoing Series will be

found to be equal to '.tW+'g > ^^^ T^in-

Let it be alfo frofofed to find the fum of the folhwit^ Seria

A B C D E F G

I r -4- 3 rr + 4 r» -4- 7 r* -h II r» -4- 18 r* -»- 19 r^ ^r.

whofe Terms are related to one another in this manner^ viz. C z^

I Br -4- 1 Arr, D = 1 Cr -4- i Brr &e. Let m And ta be re.

fpellivelj made equal to i and — I, and then that Series will be

foundequal to ' "^^^I^^ orio /j^L^.

DEMON*

^ ik^ifTJ ^-, rsTicrr:-''* r -y -> ^t.^t^r^nnt

■^FV*^^VI««lPfWn*W««W|p«*inMii^«ir^n>wv«**««. ■■ . "^.

Th Doctrine 0/C11ANCB1 is^

DEMONSTRATION.

Let the following Scheme be written dowO| viz^

A — A
B^B

CizzmBr^-^n An
D •== m Cr ^^ mBrr
E b=z rnDf^ n Crr
F = m Er -^ » Drr

&C.

This being done, if the fum of the Terms A^ B, C, D, E,
F &c. dd infinitum^ compofmg the firft Column, be fuppofed
equal to x, then the fum of the Terms of the other two Co-
lumns will be found thus : By Hypothefis^ -rf + B-4-C4-I>4-fi
&c- = jf, or J5 + C -H D -h £ &c = x — -rf; and Multi-
plying both fides of this Equation by m r, it will follow that
tn Br -4- mCr^ m Dr -^mEr&LC is -= mr x^^mr A.
Again, adding ^ + fi on both fides, we {hall have the fum
of the Terms of the fecond Column, viz. A -4- ^ -4- w JBr
-4- j»Cr-4-»fDr&c equal to ^ -4- B -hmrx ^^mr A. The
fum of the Terms of the third Column will be found by bare
inipeftion to be — nrrx. But the fum of the Terms contaln'd
in the firft Column, is equal to the other two fums contained
in the other two Columns. Wherefore the following Equa*
tion will be had, 1//^ xz=. A + B-^-mrx -— mrA^^nrrx^
from whence it follows that the value of x, or the fum of all
the Terms ^-Hfi-f-C-f-D^-fi&c. will be equal to

^ + B -^m r A

Secondly^ Let the Relation of each Term to the three next
preceding ones be exprcfTed as follows, viz. let D be = w Cr
^^a£rr -hp Ar\ and let E likewife be = mDr — n Crr
-h pBr^f and fo on: Then will the fum of aU the Terms
^ -4- 5 4- C -4- D -4- £ &c. 4^ infinitum^ be equal to

ji'\rB'\'C^mrA'\'nffA
— iwf B

Ll T*

j w>*gir>wi8t^www

. ■■■ j i ■■■> i- i ■ jw w i ppiwp;wp!<g ipjtjini i u,' '9 t ^^mir^ \m^K^mr^ m\'^ ^

t^o The Doctrine e/CHANcrr

To apply this Theorem, let it be propofed to find the fum •
of the following Series,

A B C D E r G

4r-4-i6rr-*- }6 r'-f-64r*4- loor^-f- 144 r*M- ip^r^&G
vi'hofe Numerical Quantities are related to one another accor-
ding to the Index 3 — J "^ '». correrponding xom — n -^f*
Let therefore 3, J, t be fubftitutcd in the room of m^9,fi
let alfo 4 r, — 1 6 rr, 3 6 r' be fubftituted in the room of ^, B^ C:
Then the fum of the Terms of the foregoing Series will-

be found equal to ^''^'^'^'^\^l\';i\lZi"''^ "^> or

And in like manner the fum of the Terms of the fblfowiog
Series, vizf.

r -4^ 1 rr -f- fr» -4- 20 r* -4- 7^^ + ^tfi r*^-4- 947'^ &a whole
Numerical Quantities are related to one another according ta
the Index 3 -♦- x -4- i, will by a proper fubftitutioo, be found

to be equal to .-^L-J^ ,,

Thirdly^ Let the Relation of each Term of a Series to four
of the preceding Terms be expreffed by means of the Indesi'.
m — ^ + ? *~ f > and the Sum of that Series will be-:

— wrB •(• mrr B'

— 'wrC

1— wr+nrr — /r>-J-fr*

Pourthlj^ The Law of the continuation of thefe Tfieorems*
being manifeft, they may be all eafily comprehended^ uoder/
one general Rule.

. Pifi^fyi If (he correfponding T^rms of any two^ or more-
Series, generated after the manner which we have above de*-
fcribed, be multiplyed by one another, the new Series refult»
ing from that multiplication, will alfo be exaClly fummable r.
Thusy taking the two following Series^ w«»

rH-xrr-4- 3r» +51^+ 8r»+i3[i^8te

r 4- 3rr-f-4ri+7r* + xx,r;+i8r;;att..iabotfiof

whidi^

ftmt'nm n w " i u ^ ijn ^ .w n n i i ■ ■ i w'tfmtmm^imim^^iffm'mtifimmarmim^'mmmf^

The DOCTRIKB i/CkANCBSL IJ¥'

Mfhich each Numerical Quantity is the fum oftbe two precediag
ones ; the Scries refulting from the multiplication of the cor«
refpoDding Terms will be

in which each Numerical Quantity being related to the threo'
preceding ones, according to the Index x + x — t, the fum

of that Series will be found to be = ,^ + v^- ar^;^.^

as will appear, if in the room of w — » -4- ^ there be fubftir
tuted X + X — I , and rr be written infteaa of r.

When the Numerical Quantities belonging to the Terms of
any Series are reftrained to have their laft differences eqtial
to Nothing, then may the fums of thoCb Series be alfo found
by the following elegant Theorem, which has been commu*
nicated to me by Mr. de Monmoru

Let ^r be the fir ft Term of the Sdies, and let the firft,
fccond and third diSerences, &c. of the Numerical Quantities
belonging to the Terms of the Series, be refpe£tively equal
to d! i'\ d"\ &c Then will the fum of the Series be equal

Thus, if it were propofed to find by this Theorem the
fum of the following Scries, w*.

jir -H i6rr -4- jdr' h- 6^r^-{- loor^ &a

It is plain that in this cafe A is :=r 4, i^ = ix, i<^ r= 8^
d^^^ = o ; and therefore that the fum of this Series is equal to

.^+ -^^^ + -5^7, whichisnMlucedto.if^^

REMARK.

OUr Method of fumming up all the Terms which in thele
Series are related to one another according to conftanr
Indices, may be extended co the finding of the fum of any
determinate number of thofe Terms, Thus, if J^ B, C, D
be the firft Terms of a Series, and V^ JT, T, ^ be the laft^
then will the (um of tbife Series be

A —

■■»\r».»«-"" v i I i in i i w ^ i iii I ■ » ■ . ■. ! — i M w I II . ' i ^ w.m^J***^ " I I w^ umijj i 'ijriH

i^a The Doctrine (/CiiANCss.

And if a general Theorem were defircd, it might caGly be
formed from the infpeftion of the foregoing.

Thcfe Theorems are very ufeful for fumming ijp readily
thofe Scries which exprefs the Probability of the Plays being
Ended in a given number of Games. For example, fuppofe
it be required to find what Probability there is, that in Four
and twenty Games, either A (hall win Four Stakes of B, or
JBFour Stakes of .^. The Series exprefling that Probability is,
from our XXXIV/A Problem

or, fuppofing an equality of Skill between the two Gaaie.

fters. -I- V i+-f + -l^ + -^+ TTT &C. which
ought to be continued to eleven Terms independently from
the common Multiplicator. Let this Series, whofe Terms are
related according to the Index 4 — x, be compared with the
Theorem, making A— 1^ B — -J- = x, w = 4, » = x.

and negleflir.g the Terms.C, D, V^ Xj the fum of the afore-
faid Series will be found := 8 -4- T — 7 ^; which being mul.
tiplied by the common Multiplicator ^ prefixt to it, the

Probability required will be exprcffcd by i + -r-3^— -^^
Wherefore nothing remains to be done but to find the two
laft Terms Tand ^: But thofe two Terms, by our XXXIV/A.
Problem, will be found ro be ^f7^ and -iU2«£l., ^

0.X90X, and 0.1477 nearly; which numbers bein§ fubftituted
refpe£tivelj^ in the room of T and Z^ the Probability required
will be found to be equal to a8i93 nearly. Let now this
laft number be fubtra£led from Unity, and the remainder
i>cing 0.1807, it follows, that *tis a Wager of 8x to i8» or
of 4i to 9 nearly, that in Twenty four Games or fooner, either
J (hall win four Stakes of A, or B four Stakes of A,

If

■»■ — ■ ■■ ■ II J I W ^ P u rn W ■ *■■■ >■■ ■ M l., m Biw ,11^, ^,. 1 ,, ^ . .. ,. ^ , iiy

t.>

The Doctrine 0/ Chances. 133

If the number of Stakes were Five, the fum of the Terms
of the Scries belonging to that Cafe would alfo be cxpreft by
means of the two laft Terms, fuppofing any given number of
Games, or any proportion of Skill. If the number of Stakes
were Six or Seven, the fum of the Series belonging to thofe
Cafes would be expreft by means of the three lall Terms;
If Eight or Nine, by means of the Four lad Terms, and fo
on.

LEMMA III.

IF there be a Series of Numbers^ ^i A, B, C, D, E &C. irA^
Relation is expreft by a9ij confiunt InAex^ and there be Mother
Series of U umbers^ as P, Q^, R, S, T Sjc. whoje hft Differences
Are equal to toothing ; And eAch Term of the firft Series be Mnlti*
fliei bj eAch correfponding Term of the fecond^ I fAj thAt the Pro^
diiits AP, BQ., CR, DS, ET &c. conflitute a Series cfTerms^
fphofe Re I At ion mtcf be expreft by a conftAnt Index. Thus ^ we
take the Series i, x, 8, i8, lOO &c. rvhofe Terms are reUtedfy
she Index 3-4-x» And eAcb Term cf thAt Series be refpeifrt/eJf
Multiplied bj the correfponding Terms of An Arithmetic Progre/*
fion^ fuch AS 1, 3, f, 7, 9 &c. whofe laft Differences are eqttAi
so Nothing: Then it tpill be found thAt the ProduSs I9 6, 40^
196, 900 &c. conflitute A Series of Numbers^ eAch Term of which
is ReUted to the preceding ones According to the Index 6— 5 —
I X — 4. tJow the Rule for finding the Index cf this RelAiiose
is AS follows.

Takc the Index which expreffes the ReUtion of the Terms im

the firft Series^ And Multiply cAch Term of it by the correfponding

Terms of the Liter aI Progreffion r, rr, r' &C. which being Jone^

fubtrAll the fum of thffe ProduSs from Unity ; then let the remAsn^

der be rAifed to its Square^ if the fecond Series be compofed of

Terms in Arithmetic Progreffson; or to its Cubcj if it be compofed

of Terms whofe third Differences Are ejuAl to Nothing ; or to its

fourth Power J if it be compofed of Terms whofe fourth Differences

Are equal to Nothing ; And fo on. Let thAt Power be JubtrAited

from Vnity^ And the remainder ^ hAving CAncelled the Letter r, will

be the Index required. Thus in the foregoing Example^ hAving tA^

ken the Index }-+-x, tvhich belongs to the prjf Series ^ And Malti^

plied its Terms by r And rr refpeiltvelj^ let the ProduH jr -4- xrr

Mm bt

'•^••'^^

•• I ' . 1 ■■■ j u j iii ■i nm a t ii L-— I * " *T'^ \ ,'*';- *^ "\ ' " ^ ] *' J l\^. ' * t '' '^'''''*' ^~^X^

134 ^^ Doctrine 0/ Chances.

be fuhrailed from Vaify^ dni the SquAre of the remainder being.
I — 6 r 4- y rr + ix r* -4- 4 r% /e^ that Square be alfo fubtraclei
from Vniij^ then the rem/dnder^ having cancelled the Letter r, mB
be 6 — 5 — 12 — 4, which is the Index required^

But in cafe neither of the tivo firjl Series have any of their
lajl Differences equal to Nothings yet if in both of them the Rela^
tion of their Terms be expreffed by conftant Indices^ the third Sc^
ries^ ref tilting ffom the Multiplication of the eorrefponding Terms
of the two firfl Series, tviU aljo have its Terms related to one ano^
ther according to a conflant Index. Thus, taking the Series 1,3,
y, 11, 21, 43 &c. the Relation of tvhofe Terms is exPreffed ij
means of the Index 1 4-x, and Multiplying its Terms ij the cor*
refponding Terms of the Series 1, x, Ji i)» 34^ 89 &C. the Rela*
tion of whoje Terms is expreffed by the Index 3 — i, the Produ0s
tpm compofe the Series !> ^9 15» 14)9 7i4f '^^^7$ fvhofe Terms are
Related to one another according to the Index 3 -f- 13— 6 — 4*

Generally » If the Index exprejpng the Relation of the Terms irs^
the frjl Series be m4-n, and the Index exprefpng the Relation of
the Terms in the fecond Series 4^ p-4-q ; then will the Index ef the
Relation, in the Series refulting from the Multiplication of the cw*
refponding Terms of the Two prjl Series^ be expreffed by the foU
lowing fyantities^

-4-niinq'
viz. mp -4- n p p -4- mnpq — oaqq;

4- xnq

But if it fo Happen that p be equal to m^ and q to ni then the
foregoing Theorem may be controlled, and the Index of the Rela^

tion may be expreffed as follows, via mm + nimn _^,

fo that the Relation of each Term to the preceding ones need not bi
extended, in this Cafe, to any more than three Termt.

And in like manner other Theorems may be found, which m/tf
hi extended farther^ and at lafi be comprized tender am getteral
Rule.

PRO

^, w. ■■■■■I .. I ■ ■,.- — — ■■ ■ iiii i ■ I J ■ I " ■ ■■ ■■■■ ' "

The Doctrine 0/ Changes: 135

PROBLEM XLIIL

SVffofmg A dnJ B, tvhofe proportion of SkiB is as z to b,
to Plaj together^ till A either wins tie number q of Stakes^
or lofes the number p of them ; And that B Sets at every Game the
fum G to the fum L : It is required to find the Advantage^ or
Difadvdntage of A*

SOLUTION.

Flrjf^ Let the number of Stakes to be won or loft on ei-
ther fide be equal, and let that number be ^ ; let there
be alfo an ecjuality of Skill between the . Gamefters : Then
I fay, the gam of ^4 will be pp x-^ii^ that iS| the Square

of the number of Stakes which either Gamefter is to win
or lofe, Multiplied by one half of the Diderence of the va*
lue of the Stakes. Thus, if A and B play till fuch time
as Ten Stakes are won or loft, and B Setts a Guinea to
Twenty Shillings ; then the gain of ji will be a hundred
times half the Difference between a Guinea and Twenty
Shillings, viz. 3 7 — iy«wi.

Secondly^ Let the number of Stakes be uneaual, fo that A
be obliged either to win the number q of Stakes, or to lofe
the number p ; let there be alfo an equality of Chance be-
tween A and B : Then I fay, that the gain of A will be
pqx -—^i that is, the Produft of the two numbers of Stakes,

and one half of the Difference of the value of the Stakes Mul-
tiplied together. Thus, if A and B play together till fuch
time as either A wins Eight Stakes, or lofes Twelve ; then
the Gain of A will be the Produft of the Three numbers
8, 12, 9, which makes 864 pence, or 3/ — xi****
' Thtrdljij Let the number of Stakes be equal, but let the
number of Chances to win a Game, or the Skill of the Game-
fters be unequal, in the proportion of 4 to b. Then I lay.

that the gain of A wUl be ±^^ZltLx j£=±

° A^-\-b^ 4 — *

dG~bL

T^r X-

fottrthlj^

' " "'*' ■■■ ■ ■ ■ ■ i^wp^wipiMtw^yqwwwiw^WiewwwtpiBBWWitPgff^

i^6 The Doctrine of Chances.

Fottrtblj^ Let the number of Stakes be unequal, and let al-
fo the number of Chances be unequal : Then I fay that the

gain of ^ will be ^^^>^^^-^^ -/>^^v^^^%.±!g=:i^

DExMONSTRATION.

]N order to form a general Demonftration of thefe Rules,
let us refolvc fome particular Cafes of this Problem, and
examine the procefs of their Solution : Let it therefore be
propofed to find the gain of A in the Cafe of Four Stakes
to be won or lod on either (ide, and of an cauality of Chance
between A and B to win a Game. There being an equality
of Chance for A^ every Game he plays, to win G or to lofc
L, it follows, that the gain of every Game he plays is to be
reputed to be ^'^■. But it being uncertain whether any

more Games than Four will be play'd, it follows, that the
gain of the Tenth Game, for in(lance,'to be eftimated be-
fore the play begins, cannot be reputed to be -2r£. ; for it

would only be fuch provided the Play were not Ended be^
fore that Tenth Game: Wherefore the gain of the Tenth
Game -is the Quantity -£zL Multiplied by the Probability

of the Plays not being Ended in Nine Games, or before,
for the fame reafon, the gain of the Ninth Game is the Quan-
tity J^^ Multiplied by the Probability that the Play will

not be Ended in Eight G?mes : And likewife the gain of the
Eighth Game is the Quantity .^:iL Multiplied by the Pro*

bability that the Play will not be ended in Seven Games,
and fo oa From whence it may be concluded, that the gain
of A^ to be eflimated before the Play begins, is the Quan-
tity S^ Multiplied by the fum of the Probabilities that the

Play will not be Ended in o, i, x, 3, 4, j, 6, &c. Games
di infutitum.

Let thofe Probabilities be refpeftively called A\B'j C\iy^
E\ F\ G\ &c. Then, becaufc the Probability of the Plays
not being Ended in Five Games is equal to the Probability
of its not Ending in Four, and that the Probability of its
not Ending in Seven, is equal to the Probability of its^ not

Ending

'.TiZ^'.ZT'Tr^. f^rrr ~""' ■ . '■■ ; ■ .". - ■' . ; jj. -<i n B» M ii ^ t» i i» "i^— - ■ " *■■» ■■■!■■ ^y

The Doctrine c/Chances. 137

Ending in Six, it will foHow, that the fum of the Probabilities
belonging to all the Even Games is equal to the fum of the
Probabilities belonging to all the Odd ones : We are therefore
only to find the (um of all the Even Terms, A' -^-C' ^
E' -^-G^ &c and 10 double it afterwards.

Now it will appear, from our XXXIIW Problem, tliat
thefe Terms conftitute the following Series, t//ja

"1 "t""4;*-T- 16 "t- ^ -T" aji "*• 1024 "^* 4oy*

In which Series, each Numerator being Related to the two
preceding ones according to the Index 4 — x, and each De-
nominator being a Power of 4, it follows, that this Scries ofiay
be compared with the firfl: Theorem of our feoond Lemms;
by making the firft and fecond Terms A and B^ ufed in that
Theorem, to be refpeQively equal to i and ^i^ making alio

the Quantities m^ n^ r refpeflively equal to 4, x, JL« Which

being done, it will be found that the fum of all thofe Terms
ad injimtum will be equal to 8.

We may therefore conclude that the fum of all the Terms
A' -^ B' -4-C'-HD'-4-£' & c is eq ual to x6, and that
the gain of A is equal to 16 x ~^

But if the number of Chances which A and B have to win
a Game, be in a proportion of inequality, then the fum of
the Series ^' -4- C -H iS' -f- C -f- /' &c wiO be found

thus : Let -^|> be called r, and the Terms of that Series
will be Related to one another as follows, wjc. £'= 4CV—
x^Vr, G'= 4 E'r — X CVr, and fo on. Let therefore 4,
X, I, I, be refpeftively fuWlituted, in the firft Theorem of our
fecond Lemmdyin the room of i»,/r, A\ B^i and the fum of
this Scries will be found to be —2^i^l—^. in which ex-

preflion, reftoring the value of r, w^ J^^r, the fum of the
Series will become 3iI^i2L3Hl , the double of which

is the fum of all the Terms ^' + 5' -f- C -f- D' + £' &c.
But becaufe, in eyery Game, the Gameiler A has the nqm-
ber 4 of Chances to win (7, and the number i of Chances
to lofe X } it follows, that his gain in every Game is equal
•. ^ Nn to

'» ' j T^i i w I ■■ w fm.^ m »i onrnfrnt^m"** m *jfi i

-.^xT-

138 The Doctrine o/Chances.

to -iS-pti.. From whence it may be concluded, that the
Advantage of Ay to be eftimated before the Play begins,

will be ^---^-4/j'^;::t£L X '-^^.

Before we proceed farther, we muft obferve, that the Se-
ries A' -¥ B' -^ C' -¥ D' + E' -^ F' &c. which we havcaP
fumed to reprcfcnt in general theProbabilitiesof the Plays not
being Ended in o, i , i) 3, 4, 5 &c. Games, whether the otakes
be equal or unequal, being divided into two parts, via^ A' -\-
C'-hE' + C'&c. and B' -^ D' -{- F' '\' H' &c. anfwer-
ing to o, 2, 4, 6 &c. and i, 3, y, 7, &c. each Term of thefe
two new Series will be related to the preceding ones, accor-
ding to the fame Law of Relation, as are the Terms of thofe
Series which exprefs the Probabilities of the Plays being End-
ed in a certain number of Games, under the like circum-
fiances of Stakes to be won or Lo(l. The Law of which
Relation is to be deduced from our XXXIY/iEr^ and XXXIXrA
Problems.

If the number of Stakes to be won or loft on either fide
be equal to Six, and the proportion of Chances to win a
fingle Game be as 4 to ^ ; then the Relation of each Term
to the precedinjg ones, in the Series ^' -4- C -f- £' -4- G'&c,
will be exprelled by the Index S^--^ -^ %. Wherefore to
find the fum of thefe Terms, lee the Quantities 6, 9, x, i^
I, I, be refpe£lively fubftituted, in the third Theorem of
our fecond Lemmas m the room of m^n^f^ A\ B\ C\ and
the fum of thofe Terms will be found to be

preffion fubftituting -j^^ in the room of r, the fame will
become i*''-t-v**y-^?»*^''"4^' . From whence we may
conclude^ that the gain of A will be

Again,

nny^^iiii^ ■ 11^ ^wpiwi^»Tw»iyi>.'W>— i»^>yw»yi»y^l»''^>" n i ■■ iw^i^— w»— ^m^w^^w i w iw n ■ " » "

I

\

— ^.v* «. .^MiJa

The Doctrine o/Chances. 135?

Again, if the number of Stakes to be won or loft on d*
thcr fide be Eight, ic will be found, that the gain of A will

oe X -^^-, :l

But the Numerators of the foregoing Fraftions being in
Geometric Progreflion, if thofe Progremons be fummed ujr,
the gain of A^ in the Cafe of Four Sukcs to be won or loft,
may be exprelTed as follows.

viz. by the Fraaion ^"^^"'''^1 x ^^T^^ j or

dividing both Numerator and Denominator by a+7^\ the
fame may be expreft by the FraAion ^^^T^^ x ^^2^^*
His gain likewife, in the Cafe of Six Stakes to be woo or
loft, will be expreft by the Fraftion ^^/-"^{l x ^^Z.^^'\2xA
in the Cafe of Eight Stakes to be won or loft, it will be ex«
prcft by the Fraftion ^^/f"//' x ^^^-^^ : So that wc

may conclude, that in any Cafe of an Even and equal num-
ber of Stakes denominated by f , the gain of A will be ex-

p—vbP

preft by the Fraftion ^"^"^^ x

mG — it

^-T •

But if the number of Stakes be Odd and equal, as it is
in the Cafe of Five Stakes to be won or loft, then the two
Series ^'4-C'-h£' -4-6' -4-/' &c. and B^-h D'^a'
&c. will be unequal, and the excefs of the firft above the
fecond will be Unity. Wherefore to find the gain of A^ in
the Cafe of Five Stakes, having fet afide the nrft Term of
the firft Scries, let all other the Terms be added together,
by comparing them with thofe that are employed m the
firft Theorem of our fecond Lemmd ; which will be done
thus. Since C' = x, £' = i, and G' = f E^r — $ C rr^
let the numbers i, i, 5, 5, be refpeftively fubftituted in the
aforefaid Theorem, in the room of the Letters A'^ Bf^m^ »\

and

■ ■ IfW ■ !■ — P^^P^KWI LUfJ l f I"! ' ^1 ■ ! IIW L il ^ P H. I . y H^^tf Ti

140 . T^tf Doctrine o/Chances.

and tlic fum of that Series will be found to be - — ^-~t -,r-^*

To ilie double of which adding Unity, which wc had fct
afide, it will appear that the fum of the two Series together

will be -J-E-i7^77> ^^ writing -^^jr in the room of r,

■ ^''t^'^^^!L'^^^^^^^^xj^^*^ now by rcafon that the

Terms of both Numerator and Denominator of this laft
Fradion compofe a Geometrick Progreflion, the Numerator

will be reduced to -JLlL^JLil.^ and the Denominator will
be reduced to -^^^7^. From whence it follows, that the
fum of thcfe two Series will be ^l^^IUl^EL, and that

the gain of ^ will be y;;-T{' x ~=i^' ^^*^

gain of ^ be likewife inquired into, in the Cafe of Seven
Stakes to be won or loft, then ic will be found to be

And the fame form of expreffion

^' + ^' ^ #-^

being conftantly obferved in all cafes wherein the number
of Stakes is Odd and eaual, we may conclude that if that
number be denominated by f^ then the gain of. A will be

P^^-V^J X f'^-^^ , Now this exprcffion of the

gain of A having been found to be the fame in the Cafe
of an Even number of Stakes, as it is now found in the
Cafe of an Odd one ; we may conclude, that it is general,
and belongs to any equal number of Stakes whether Even
or Odd.

If the number of Stakes be unequal, the Inveft^ation of the
gain of ^ will be made in the fame manner asit was in the
Cafe of an equality of Stakes. Thus, let us fuppofe that the
Play be to continue till fuch time as either A wins Two
Stakes, or B Three. In order therefore to find the. gain of
.A^ let the Scries ^' -:*- 5' + C' -H D' -♦- £' -f- F^ &c.

be

^B*W^MPW^r*'*"^^'^^^»"'— ^^|^^^^»^^*»<^1KP^»'—Wii^f^^^^^t— ^IPW— -—— ^—p^— ^— ^»«l^— ^»^»^w» ■ ■ I 1

TJ)e Doctrine ^Chances. 141

be divided into two parts, viz. ^' -4- c' -4- iS' ice. and
fi' 4- £>' H- f ' &c then it will appear, from our XXXIIW,

and XXXIX/A Problems, that A' = x, C = ^^*+/*

E' '=z i C'r — 1 A' rr. Having now obtained the firft
Terms of the Series, and the Relation of each Term of it
to the preceding ones ; it will be eade to find the fum of all
its Terms, by the help of the firft Theorem of our fecond
Lemmsy making the Quantities, Aj 5, m^ n therein employed

to be refpeftively equal to i, -y—ji^f 3, u This donc^

the fum of that Series will be found to be equal to

that in the fecond Series 5' is r= i, D'= -H«.» + |f**.»
T' z=z iD'r — B' rr\ from whence the fum of all its Terms

wiU be found to be /l"^, Vt "^ ?/ V^ ^t jT i4 ■: And both
fums of thofe Series being added together, the aggregate of
them win be ^'l t y/ j ^^^^^ W'l JT^- But the

Terms of this Denominator compoflng a Geometric Progref«
fion, whofe fum is ^^^"^^^ , the foregoing Fraction may

be reduced to i^^-f »^>^n i^^^^ail^^^T^xrfT, ^jjjch

Fraflion is dill capable of a farther redu^ion ; for the three
firft Terms of its Numerator compofe a Geometric Progreffi-
on, and the two laft Terms may be confidered as being in
Geometric Fro^reflion, and confequcntly the Fra£lioa may

at laft be reduced to "^^ .TZ1,^3^,,,-737^x.1:T ^
from which expreffion, the gain of A will be found. to be

Oo By

ifwii ■■ ■ w » ^1 p . m 1 1 •rrm^if^mrr'mr^mmr'mrmtmmm' . » ■ w »-i"<>w<wr^>-^w>n^>>w^— <iy

14^ fhe DoGTRiNB 0/ Changes.

By the fame method of Procefs, it will be eafy to deter-
mine the gain of A under any other circumilance of Stakes
to be won or loft : And if ic be remembred always to fund
up thofe Terms wliich are in Geometric Progreflion, all the
¥arious exp! cdions of the gain of J^ calculated for ditfcring
numbers ol Stakes, will appear to be uniform : From whence
it may be collcfled by bare infpedion, that the gain of A is
what wc have alTcrted it to be, viz.

It is to be obferved, Firfi^ that if ^ and q be equal, the fore-
going cxpreffion maybe reduced to ^^^,T^/ - x - _. ^

as will appear if both Numerator and Denominator be divi-
ded by 4^ — h ^ having firft fubft.tuted f in the room of jr.
SeconJtjy that if 4 and b be equal, the fame exprefllion may

be reduced lopqx -^^> which will appe ar if both Numc- -
rator and Denominator be divided by a — 6\

After I had Solved the foregoing Problem, I wrote word ■
of it to Mr. Nicolas Bernoullj^ the prefent Piofeflbur of Ma*
thematics at PaJIoua^ without acquainting him with my So--
lution : I only Kt him know in general that it was done by
the Method of Infinite Series ; whereupon he fcnt me two
different Solutions of that Problem : And as one of them has
fome Afi^nity with the Method of Series ufed all along ia
this Book, I (hall tranfcribe it here in the Words of his Let-
ter, ^* My Uncle has obferved that this Problem may alio
^^ be Solved after the fame manner as you have Solved the
** Ninth Problem * of your Traft de Menfurd Sortis^ it be-
^ ing vifible that the Expe£tations of the Gamefters will re-
*' ceive no alteration whether it be fuppofed that the Pieces
•* which A and B Set every time to each other, are refpeflive-
*' ly L and G, or whether it be fuppofed that thofe Pieces
** conftiiute the following Progreflion, n/U.

yG-^L &c. the number of whole Verms is ^ + f , whereof

! ^ tit ULih tmkm ti tku ^if^

the

9IPV*«p!i^4«a^W«W*BM-a^^'^IMM»WI^>^MI*Ml'^N«f^P«M^^jgyf«0^iW^«>w*w^»MOT^Havi^^^f«»V>Ba h ii^m^^t^fm

\

I

\

The DoCfkiNE^CHANCBS. 143

•

^^ the firft, whofe number is /, denote the Pieces of A\ and
** tile lally whofe number is ; , denote the Pieces of B : For 10

^ either Cafe the gain of A will be J *^;^^^ - . Now it being
'* polTible to find the fum of any number of Terms of thisPro*
•* grcffion, it follows that the different values of all the Pieces
*^ of each Gameller may be obtained: Let therefore thofe
^' values be denoted relpedively by 5 and T; let alfo the
•* Probabilities of winning the numoer of Stakes agreed up-
^ on be called A and B refpeflively, which Probabilities arc

aP^i^^irbTn- ^°^ a^i^i^^f^ ^"**^ ^ ^^^ feveraUy

** derived them, your felf in your aforefaid Problem, and I
" in Mr. fltofftmors\Book. This being fuppofed , the gain of A

^ will be found to be AT^BS, or -r f^j/^ x ^ i^Bf.

It. B. Tho* I may, accidentally, have given a ufeful Hint
for that elegant Method of folving the foregoing Problem, yet
I think it reafonable toafcribe it entirely to its proper Author ;
the Hint having beea improvM much beyond what I could
have expected, ^

RE MAR IC.

r' is to be obferved, that the gain of A is not to be regu*
lated by the equal Probability there is that the Play may,
or may not be E'ided in a ccnaiu number of Games. For
inftance. If two Gameflers having the fame number of Chan-
ces to win a Game, dcfigi only to play untill fuch time only
as two Stakes are won or loft ; it is as Probable that the Play
may be Ended in two Games as not, yet it cannot be con^
eluded from thence, that the gain of y^ is to be eftimated
by the Produfl of the Number % by one half of the DiSe-
rence of the Stakes : For it has been Demonftrated that this
gain will be Four times that half difference. In like manner^
if the Play were to continue, till either A fhould win Two
Stages, or B Three ; it will be found, that it is as Probable
tint th;: Play may End in Four Games as not; and yet the
g*i:i of J i> n it to be eftimated by the Froduft of the Num-
ber 4 by one half of the Difference of the Stakes} it ha«

VlDg

:>

■ I ■ Mii—^w w i ym I wmt m ^vmn^fnafmrwmwn m uii iw i 'li iW^WP*''^^>^W"**''— *^?**Wf*W»

£44 ^ DocTiiiNE of Chances.

ving been Demonftrated that it is Six times that half DifTo
rence. To make this the more fenfible, let us Tuppofe that
A and B are to Play till fuch time as A either wins one
Stake, or lofes Ten: It is plain, that in this Cafe it is as Pro-
bable that the Play may be Ended in One Game as not, and
yet the gain of A will be found to be Ten times the Diffe-
rence of the Stakes. From hence it is plain, that thb gaia
is not to be edimated, by the equal Probability of the Plays
Ending or not Ending in a certain number of Games, but
by the Rules which have been prefcribcd in this Problem.

PROBLEM XLIV.

IF A AnA By vshoft froportion of Skill is ^r/ a /d h, refolvifig
to PlMjf together till fuch time as Four Stdkes are worn or lyt
on either Jide^ Mgrre between themfelves^ that the Srft Game thst
is plsfdy they JbtU Set to each other the refpellive Jams L asfJ G ;
thst the fecond Game they Jhdll Set the turns x L amd xG\ the
third Game the fums ) L and } G« and jo on ; the St dies inerea^
fing eontinudHj in an Arithmetic Progreffion •• // is Demanded bom
the gain tf iL k to he ejl mated in this Cafe^ hefon the Play
hegttss.

SOLUTION.

LET there be fuppofed a Time wherein the Number jp
of Games has tKen playM ; then A having the Num-
ber a of Chances to win the fum |^+ i x G in the next Game,
and B having the Number h ot Chances to win the fum
f -l-ixL} it IS plain, th at the gai n of A in that drcumftance

of Time will be p+ix ^^^\^ ' . But this gain being to be
eftimated before the Play bcgms, it follows, that it ought to
be eftimated by the Qiiantity p + i x ^^T f - multiplied by

the refpeAive Probability there is that the Play will not
then be Ended ; and therefore the whole gain of A is the
fum of the Probabilities of the Plays not Ending in o, i, x,
Jf 4i 5f 6 ^c* Games ad infinitum^ multiplied by the rcfpeft.
ive values of the Quantity ^>t>ix^^^T^^> f being Inter-

preted

Wt^iwiy^rw;^— w ^^i l i

'^Wy^WffW^pfl^^fi^^i^liflpiiWWO^WWigpM^i^^.ii ^ t f i. W n |in II im ^my

The Doctrine q^ Chances: 14?

•prcted fucccffively by the Terms of the Arithmetic Progref-
fion, o, r, X, 3, 4, f, 6 &c. Now let thcfc Probabilities of
the Plays nDt Ending be refpcftively called ^', B\ C D',
E\ F\ G' &c Let alfo the Quantity ^^—^^^ . be called 5;

and thence it will follow, that the gain of ^ will be A^S
-4-25'5 -4-3C'5 -4-4D'5 + y £'5-»-6/''S&c Butia
the Cafe of this Problem B' is equal to A\ and D' is equal
to Q\ and fo on. Wherefore the gain of A may be expref*

fed by the Series S x 3^' + 7C' + \\E' + 15^'+ ijr'&c.
But it appears, by our XXXIIli Problem) that the Terms
A\ C' E'j G' are refpeftively equal to the following

auantities, viz. i, i, A-^h ^r 6^bb + ^abi ^

the Terms 3 -4' -4- 7 C -4- 11 £' -4- 15 G' may be obtain-
ed: It appears alfo, from what we have obferved in the
preceding Problem, that the Relation of the Terms A\ C%
E' &c. may be exprelTed by the Index 4 — %\ and by the
Third Lttmrns prefixt to that Problem, that the Relation of
the Terms 3 -4', 7C', 11 £' &c. may be expreffed by the
Index 8— xo-4-16 — 4: And therefore fubftituting the
Quantities 3 ^', 7 C, 1 1 £', i y G ' in the room of the Quanti-
ties A J Bj Cy Df which we make ufe of in the Third Theo-
rem of our fecond Lemma ; fubftituting likewife the Quan-
tities, 8, 2o, 1 6, 4 in the room of my n^f^ji .and laAlyTub-

fiituting *[^^^ in the room of r ; the gain of A will be
expreft by the following Quantities, viz.

which, in the Cafe of an equal number of Chances to win a
Stake, would be reduced to x 1 6 6' ; and therefore if the Quan-
tities G and L (land refpeflively for a Guinea and Twenty
Shillings, which will make the value of 6' to be Nine pence,
it follows, that the gain of ^ will in this Cafe be 8 / — x ^'•

P p CeroBdrj L

'^*^*''*™"' ' ■ ■ n^yw. w w i w IIP » I i wwpwuiiw^i ■■■! i«" .t i » i< y> i " n

14^ ^he Doctrine 0/ Chances.

Corollary I. If the Stakes were to Increafe according to the
proportion of the Terms of any of thofe Scries which we
have defcribcd in our Ltmmtfs^ and that there were any gi. .
ven inequality in the number of Stakes to be won or lod, the
gain of A might Aill be found*

Coroltary II. There are fome Cafes wherein the gain of A -
would be Infinite : Thus, if A and B were to Play rill fuch
times as Four Stakes were won or loftj and it were agreed be- •
tween them to double their Stakes at every Game, the gain
of A would in this Cafe be Infinite: Which confequence .
may eafily be deduced from what has been faid before.

PROBLEM XLV.

IF A And B refolve to PUj tiB fueb time as A either wins
d cert Ain given number of Sfdkes^ or that B tpins the fdme^ •
or fome other given numher of them •• ^Tis required to find in
how many Gdmes it tpill be as Probable that the Play may be-
Ended as notf

solution:

LE T it be fuppofed that A and B are to play till fuch •
time as either of them wins Three Stak^ and that
there is an Equality of Skill between them. This being
fuppofed, it will appear, from our XXXI V/A Problem, that
the Probability of the Plays continuing for an Indeterminate
number of Games may be expreft by the foHowing Series^

•"r?Tn"xi + 7pr- + -|^ + -^^^ Which, m

the Cafe of an Ec]uah*ty of Skill between the Gamefter^ wiD
be reduced to this Series,

III ▼ vn nc ^
^X i-f--i- + -^ + -2L.&c.whofcTcrmsarercfpeai7c-

ly correfponding to the number of Games 3, y, 7 9 ice
Wherefore fo many of thofe Terms ought to be taken, as
that their fum being multiplied by the common Multiplica-

tor

9!

Sgq8iWiSf '^>ijj.%jrii ! frM i i*»* i ^W '«g^MfWi iw * » ' W| i .w wwf^piw*iiWi^^^^ ■ *^

L.,

The DocTKiNB ^Chances. 147

tor -J- or -1-, the Produft may be equal to the Fradion -L.,

which Fraftion denotes the equal Probability of an Events
Happening or not Happening : But if two of tfiofc Terms be
taken, and that their fum be Multiplied by -i-, the Produft

will be -jL. ; which being left than the Fraftion -i-, it may

be concluded that Five Games are too few to make it as-
Probable that the Play will be Ended in that number of
Games as not; and that the Odds againft its Ending in Five
Games are 9 to 7. But if Three of thofc Terms be taken,
then their fum being multiplied by the common Multiplica-
tor — L-, the ProduA will be -|^ ; which exceeding the Fraft*

ion -^9 it may be concluded that Seven Games are too ma-
ny ; and that the Odds of the Play being Ended in Sevea
Games, or fooner, arc 37 to 27, or 4 to j very nearly.

tf. B. It would be needleft to inquire whether Six Games
might not bring the Play to an equal Probability of Ending
or not Ending ; it having been obfcrved before, that in the
Gafc of an equality of Stakes to be playM for, it is impolfi-
blc that the Play (hould End in an Even number of Games^
if the number of Stakes be Odd ; or that it ihould End
in an Odd number of Games, if the number of Stakes be

Even.

In like manner, if the Play were to continue till Four
Stakes be won or loft on either fide : Then taking .the fol-
lowing Series, vvc.

the fuppofition of an equality of SkUl between the Game-
Iters, may be reduced to this, vix^

vf vi vin X XII

_?_ V , . ,A- . -!A- g, -jg, ^1, '* ~ «c. let 10 many

of its Terms be tried, as will make the Produft of their fum
multiplied by -^, equal to the Fraftion -i-, or as near it
as poffible. Now Five of thofe Terms being tried, and their
fum being multiplied by -^, or -J-, the Produft will

be a|5|.^ which not differing much from -i-, it may be

€on*>

I

■ ■■^i — ] ■ Hi^pa t w ^p p^ WH llW HIl Wfi . 1 I ■■■■■■ '^ --■-' ' - ^. -^

i/^8 The Doctrine 0/ Chances,

concluded that Twelve will be very near that number of
Games, which will make the Probabilities of the Plays End-
ing or not Ending to be equal; tlie Odds for its Ending
being only 1091 to 95^6,01* 8 to 7 very nearly. But the
Odds againd its Ending in Ten Games, will be found to be
39 to 19, or 4 to 3 nearly.

By the fame method of Piocefs, it will be found that
Five Stakes will probably be won or loft in about Seven*
teen Games: It being but the Odds of 11 to 10 nearly, that
the Play will not be Ended in that number of Games, and
10 to 9 nearly, that it will be Ended in Nineteen*

It will alfo be found that Six Stakes will probably be won
or loft in about Twenty Six Games, there being but the
Odds of 168 to 167 nearly, that the Play will not be End-
ed in that number of Games, and i; to xx nearly, that
it will be Ended in Twenty Eight.

If the fame Method of Trial be applied to any other num-
ber of Stakes, whether equal or unequal, and to any pro*
gortion of Skill, the number of Games required will always
e found.

Yet if the number of Stakes were great, thofe Trials
would become tedious, notwithftanding the Help that might
be derived from our Second Ltmrns^ whereby any numoer
of Terms of thofe Series which are employed in the Solution
Of this Problem, may be added together. For which reafoa
It will be convenient to make fome Trials of another nature,
and to fee whether, from the refolution of ibme of the fim-
fjeft Cafes of this Problem, any Analogy can be obferved
between the number of Stakes -given, and the nuniber of
Games which determine the equal Probability of the Plays
Ending or not Ending.

Now Mr de Monmort having with great Sagacity dilco-
vered that Analogy, in the Cale of an e^ual and Odd num-
ber of Stakes, on (uppofttion of an equahty of ^kill between
the Gamefters, I thought the Reader would be well pleafed
to be acquainted with the Rule which he has given for that
purpofe, and which is as follows.

Let n be any Odd number of Stakes to be won or loft
on either fide ; let alfo ^^^ be made equal to f: Then the

Quantity 3^^ — 3?+i will denote a number of Gamers,

wherein

^,4n7CiS;rp:Aj-T~. ■ tie .' zi^pzT'. ■ ^ •.••f^' ■>ni.w<]i"'wjppip— ^i^aMwyj^piw^^iiwimpwpiww— p— wwww

TiW mt

\

ne Doctrine 0/ Chances. 149

wherein it will be more than an equal Chance that the Vhy
will be Ended ; thus, if the number of Stakes be Nineteen,
then f will be 10, and the Quantity ^ff^^^pj^ i will
be X7I, which (hews that 'tis more than an equal Chance
that the Play will be Ended in xyi Games.

The Author of this Rule owns that he has not been able
to find another like it, for an Even number of Sukcs ; bus
I am of opinion, that tho* the fame Rule, being applied to
that Cafe, may not find the juft number of Games wherein
there will be more than an equal Probability of the Plays
Ending, yet it will always find a number of Games, where-
in it is very near an equal Wager that the Play will be
Ended. Wherefore to make the Rule as extenfive as it may
be, I would Chufe to exprefs it bjr the number of Stakes
whether Even or Odd^ and make it -^»fff which difK:rs

from his own^ but by the fnull FraQioa .^^

If any one has a mind to carry this Q>ecuIation dill farthert
and to try whether fome general Rule maf not be di(cover«
ed for determining, by a very near approximation, the num-
ber of Games requiHte to make it a Wager of any given
Eroportion of Odds, that the Play will be Ended in that num*
er of Games, whether the Skill of the Gamefters be equal
or unequal ; let him Solve feveral Cafes of this Problem in
the following manner, which I take to be as expeditious as
the nature of the Problem can admit of.
Upon a Diameter equal to Unity, if fo be the Skill of

the Gamefters be equal ; or to the Quantity -^ V^i -f if their

Skill be in the proportion of d to tf let a Semicircle be de-
fcribed, which divide into fo many equal parts as there are
Stakes to be won or loft on either hde^ fuppofing thofe Stakes
to be equal. From the Firft, Third, Fifth, Seventh &c.
Points of Divifion, beginning from one extremity of the Di-
ameter, let Perpendiculars fall upon that Diameter, which
by their concourfe with it, fhall determine the verfed Shes of
fo many Arcs, to be taken from the other extremity thereofl
Let the grcateft of thofe vivfti Sines be called tn^ the next
lefs /, the next to it f, the next / Sec Make alfo

<lq « ~p

■ ^ I ■■■ ■■ '■ ■■ ■ ■ ■■ !■ ■■ p^n^wt^nw^wpr^rfwei^P^

150 Ti&e Doctrine o/Chance&

10 — pxM — 9 xm— s

F — g xp— * xp — Jw

1 — i XI— WX I — p - _^ ^
^ — sxq — mxq^^p

« — tn X i — p xi — g
&C.

then will the Probability of the Play's not Ending in a number
of Games denominated by x^ be expreft by the Quantities

i»"r*^ -h p-r'iH- jrT* c + i'i"*!? &c if the
number of Stakes be Even^ or by the Quantities

m ^-i -h ^ ""a^iB + f *"*" C -»- sT^DScc if the
number of Stakes be Odd.

EXAMPLE L

LE T it be required to find what Odds there is^ diat la 40
Games there will be Four Stakes won or loft on either

fide.

Having divided the Semicircle into Four equal parts^ ac-
cording to the abovementioned directions, the Quantity m
will be the Ferfed Sine of 13^ Degrees, and the Quantity f
will be the Ferfed Sine of 45 Degrees, which by the help
of a Table of Sines will readily be found to be oAs}%S and
0.14645 refpeaivcly. Moreover the Quantity A being e.
qual to -ji^, and the Quantity B to izx, will be found

to be I.X07X and — 0.107 1. From whence it follows, that
the Probability of the Plays not Ending in Forty Games may

be expresM by the two following Produ6ls aSjajji** x 1.1071

— 0.146451*' X 0.1071, of which the Second may be en*
tirely neglected, as being inconliderably little in refpeQ of

(he

■-''• : ■'^SSJSSiVfSWSK--"?*^ -~* -J^ l UM. t mj . . ■ ■ i ju i i m m! m , » ' . wt L * tf »a.-^^ • —j* •*«• •'•----■^♦«-— — ■•— v

TJ}e Doctrine o/CnANCES. 153

Now the Corrcfponding Terms of chofc two Scries being
Multiplied together, the Produds« fuppoHng r equal to die
Fraftion -^, will compofe the following Series, viz.

XT -4- 3crr +385 r» -4- 48oor*H- f940or> &c in which
Series the Index of the Relation of each Numerical Quan*
tity to the preceding ones, may be found by the help of our
Third Lemmd : For the Index of the Relation in the Nume-
rator of the Firft Series being 4 — x, and the Index of the
Relation in the Numerator of the Second being 5 • — f,
which Relations are deduced from the XXXIV/A Problem,
it follows, that if in the Theorem of our Third LemmA^ the
Quantities 4, — x, 5, — 5, be refpeftively fubftituted in the
room of the Quantities m^ n^ p, jr, the Index of the Relation in
the Third Series will be found to be xo— iio-hxoo-^ 100;
wherefore all the Terms of this Series may be fummed up b/
the Third Theorem of our Second Lemmd^ fubftituting the
Quantities to, 110, xco, 100 in the room of the Quantities
m^ fly ff f , therein employed ; fubftituting alfo the Terms
xr, 30 rr, 385 r*, 4800 r* in the room of the Quantities
jly By C, D : For after thofe Subftitutions, the fum of the TUrd

Scries wiU be found to be . , i'::'rtl'lx. . >

1 — aor-f- no rr — aoor' -f- loor*

which is reduced to -il^ by changing the Quantity r in.

to its value -i--. Now fubtrafting the Fraftion -12£-

from Unity, the remainder will be the Fraction - ^o ,, the

Numerators of which two Fraftions expre(s the Odds of the
Firft Plays Ending before the Second, which confequently
will be as 476 to X47, or xy to 14 nearly.

If in the foregoing Problem, the Skill of the Gamefters
had been in any proportion of inequality, the Problem
might have been Solved with the fame eafe.

When in a Problem of this nature the number of Stakes
to be loft by either J or fi, does not exceed the number
Three, the Problem may be always readily Solved without
the ufe of the Theorem inferted m our Third Lemmd ; tho*
the number of Stakes between C or D be never fo great.
For which reafon, if any one lias the curioHty to try, if from
the Solution of feveral Cafes of this Problem, fome Rule may
not be difcovered for Solving the fame generally ; it will te

R r COS-

~' - ■ III! I liiBi ii n ^piiui jaiiIT T I III I IT T^T"^— "— — — ^*r*TT1D;

154 ^^^ Doctrine of Chances.

convenient he (liould compare together the different Solutions^
u'hich may refulc from the fuppofirion that the Stakes to be
loft by either ^ or J3 are Two or Tliree ; and then the Cafe
of the foregoing Problem may alfo be compared with all the
reft : Yet as tlicfc Trials might not perhaps be fufficient to
difcover any Analogy between tlioR Solutions, I have thought
fit to add a new Theorem in this place, whereby Four Cafes
more of this Problem may be Solved, 'viz.. When the num-
ber of Stakes to be loft by A or 5, and by C or D, arc
4 and 6, 4 and 7, j and 6, j and 7 ; The Theorem being
as follows.

If there be a Series of Terms whofc Relation is exprelfed
by the Index / -*- «i -h /i, and there be likewifc another Series
of Terms whofe Relation is cxpreffed by the Index ^-f-y;
and the Corrcfponding Terms of thofc two Series be Malti-
plied together : Then the Index of the Relation in the Third
Scries, refulting from the Multiplication of their corre(pond«^
ing Terms, will be expreifed by the Qjiantities.

x«r J H- Imfq 4- xlnqq

It is to be obferved, that altho' thefe forts of Theorems might
be applicable to the finding of the Relation of thofe Terms,,
which are the Produfts of tlie corrcfponding Terms of two
different Scries, both of which confift of Terms whofe laft
Differences are equal to nothing; yet there will be no necef-
fity to ufe them for that purpofe, that Relation being to be
found much Ihorter, as follows.

Let e and / denote the rank of thofe Differences which are
rcfpeSively equal to nothing in each Series; then the Quan-
tity e-^f— I will denote the rank of that Difference which
is equal to nothing, in the Series refulting from the Multipli-
cation of the corrcfponding Terms of the other two ; and con-
fequently the Relation of the Terms of this New Series will
caiily be obtained by our firft LimmM.

After

> pii w ui ■ p ^1* — ■^».yy^w»— *a^'f^— tpii—l»^i*W<tiH»^**^g^*» * " ■* ^^^■^w^ . T .!■■■■ I t » y ^ ^

363 ••

Tifie Doctrine o/Chances. 155 -

After having given the Solution of feveral forts
of Problems 9 each of them containing fome de^
gree of Difficulty not to be met with in any of the
reji } and having thereby laid a fufficient foundation
for folving the mojl intricate cafes that may cccurr in
this SubjeH of Chances^ it might almojl feem fufer^ *
Jluous to add any thing to this TraSl : Tet confider^
ing that a Variety of Examples is the properefi
means of making Rules eafy and familiar j and de^
figning to be as ufeful as foffible to tlyofe of my-
Readers^ who perhaps may not be fo well verfed in
Algebraical Calculations^ I have chofe to fill up the
remaifiing Fages of this BooJi, with fome eafy Fro^.
hlems relating to the Games which are moft in ufe^
fuch as Hazard, Whisk, Piquet, isfc^ and
to enlarge a little more upon the DoSlrine of.
Combinations.

T

PROBLEM XLVII.

find at HAZARD the AivanUge of the Setter up^
on aU Su^pofitions of Main dnd Chance.

SOLUTION.

LE T the whole Money PlayM for be conficlered as a com-
mon Stake, upon which both t!:e Setter and Cafter have
their feveral Expcaaclons ; then let thofe ExpeAations be de«^
termined in the following manner,

Firjl^ Let it be fuppofed that the Main is vil \ then if the
Chance of the Carter be vi or i//7/, it is plain that the Set-
ter having Six Chances to win and Five to lofe, his Expefta-
tion will be -^ of the Stake : But there being Ten Chances

out

■^arwT^pwwwwF* ■ "^ j— i n n ■— »inw|p^i»HPfTP— |gy^*^w^^''^ff<W"

15^ the Doctrine 0/ Chances.

out of Thirty-fix for the Chance to be vi or a////, it follows,
that the Expectation of the Setter, refulting from the Pfoba-
bility of the Chance behig vi or v///, will be ^ multiply.
cd by -i-, or J^ to be divided by jtf.

Seconal)^ If the Main being vii^ the Chance fhould Hap-
pen to be c/ or /x ; tlien the Setter having Six Chances co
win and Four to lofc, his Expcflatlon will be -£- or -i-

of the Stake : But there being Eight Chances in Thirty-fix
for the Ciiance to be v or /jc, it follows, that the Expeflation
of the Setter, refulting from the Probability of that Chance,
will be -5- multiplied by -i-, or -21. to be divided by 16.

Thirilj^ If the Main being vii^ the Chance ihould Happen
to be iv or x \ then the Setter having Six Chances to win
and Three to lofe, his ExpefUtion will be -^ or JL. of

the Stake: But there being Six Cliances out of Thirty-fix
for the Chance to be iv or jc, it follows, that the Expedatioa
of the Setter, refulting from the Probability of that Chance,
virill be ^ multiplied by -£., or 4 divided by 36.

Fourthly^ If the Main being viiy the Cafter ihould Hap-
pen to throw //, ///, or xii\ then the Expe6latioa of the Set*
ter will be the whole Stake, for which there being Four
Chances in Thirty-fix, it follows, that the Expe£latioa of
the Setter, refulting from the Pr<^ability of thofe Ofes, will
be -iL. of the Stake, or 4 divided by )6.

Lafllj^ If the Main being w, the Cafter (hould Happen
to throw vii or xi, the Setter lofes his Expefiation.

From the Solution of the foregoing particular Cafes it foN
lows, that the Main being z;//, the Expedation of the Setter will

be expreft by the following Quantities, viz. Jfe. + il + -i. + •£•

which may be reduced to -f|~. Now this FraAion being
fubtrafted from Unity, to which the whole Stake is fuppo-
fed equal, there will remain the Expe6lation of the Caller
viz, .Hi..

♦ 95

But the Probabilities of winning being always proportional
io the Eipeftations, on fuppofitioa of the Stake being fixt,
it follows, that the Probabilities of winning for the Setter

and

■— --— r Tr^ - ; ' ^V^-V^' ?»■'»■ Ji~ifiki' ? ■ V Jwi «C iTrf" ■ ■ i -^ S » r i > I'-'ya^ MM^-w^Mvcvip.i.ij III I . iHtM ■*■

■1 I w i^pi!

I

The Doctrine o/CflAwtrrs. 157

and Carter arc rcfpcftivcly Proportional to the two numbers
2f I and 144, which properly denote the Odds of winning.
Now, if we fuppofe each Slake to be i, or the whole Stake
to be X, the Gain of the Setter will be txpreft by the Fraflion
^j^, it being the Difference of the Odds divided by their Sum,
which fuppofing each Stake to be a Guinea, will be about

By the fame Method of Procefs, it will be found that the
Main being vi or viii^ the Gain of the Setter will be

which is about 6 d : -i-/ in a Guinea,

It will alfo be found that the Main being v or ix^ tbe
Gain of the Setter will be —if-, which is about AiLixJ^f

in a Gumea«

CoroU. f. If each particular Gain made by the Setter, in
the Cafe of any Mam, be reljpedtively Multiplied by the
number of Chances there are for that Main to come up»
and the Sum of the Products 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 tbe firft
Main, may be cftimated to be .^^ -|- .112|. -|- ^si^ to be

divided by 14; which being reduced will be ^ very

nearly, or about 4^. : x JL./. *

Corod. X. The Probability of no Main is to the Frobabili*
ty of a Main, as 109 +x to 109 — x, or as iii to 107.

CoroH. J. The Lofs of the Cafter's hand, if each throw
be for a Guinea, and he confine bimfelf to hold it as long as
he wins, will be -j^ or about ^L in all, the Demonftratioa

of which may be deduced from our XXIV/i& Problem.

PROBLEM XLVIIL

IF Four Gdm^ers /^/.gf 4/ W H I SK ; Whd^ sre the Odds fba
anj two of tbe Psrtntrs thdt sre pitched ufott^ have not the
four Honour si

Sf

SOLO-

g»'=^j«'?^-*i ,v-..-T4Tjgsy . i^PM ■ha».^a ^5r

158 The Doctrine 0/ Chances:

SOLUTION.

FIrfty fuppofe thofe two Partners to have the Deal, and the
laft Card which is turnM up to be. an Honour.
From the fuppofition of thcfe 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 mud be had to
the Theorem given in the Corollary of our XX/A Problem, in
which making the Quantities n^ r, </, ^, 4, refpeftively ec|ual
to the numbers 51, a;, 269 3, 3, the Probability requured

will be found to be ^^ ^ '^ ^ '? or #^*

Secondly J If the Card which is turnM up be not an Honour,
then we are to find what Probability the Dealers have, of
taking Four given Cards in Twentv five out of a Stock con«
taining Fifty one, which by the atorefaid Theorem, will be

found to be UJLl±2LlULll'or .liL..
louna ro dc j, x 50X49 X48»"*^ 4998

But the Probability of taking the Four Honours being to
be eftimated before the lad Card is turnM up ; and there
being Sixteen Chances in Fifty two, or Four in Thirteen
for an Honour to turn up^ and Nine in Thirteen againft it ;
it follows, that the Fraftion exprelfing the Probability of the
Firft Cafe ought to be Multiplied by 4 ; that the Fra£tion
expreding the Probability of the Second ought to be Multi-
plied bv 9 ; and that the fum of thofe Produfts ought to be
divided by i^; which being done, the Quotient ^9 or .JL.

nearly, will exprefs the Probability required.

Corollary^ By the help of the abovecited Theorem, the fol-
lowing Conclufions may eafily be verified.

It is 27 to X nearly that the two Dealers have not the
Four Honours.

It is 23 to E nearly that the two Eldeft have not the
Four Honours;

It is 8 to I nearly that neither one Side nor the other have
the Four Honours.

It

The Doctrine o/Chancbs. 15^

It Is 13 to 7 nearly that the two Dealers do not reckon
Honours.

It is a o to 7 nearly that the two Eldeft do not reckoa
Honours.

It is x; to 1 5 nearly that either one Suie or the other do
reckon Honours^ or that the Honours are not equally divided^

PROBLEM XLIX.

0/ RAFFLING.

IF am number of Gamejters A^ B, C, D &C flaj st Raffles:
Whdt is the Probabilitj that the firft cf them hdvpfg gffi hk
ChaHce mns the^ Money 0f the Plof f

SOLUTION.

IN order to Solve this Probleniy it is neceflary to have a
Table ready composed, of all the Chances which there
are in three Raffles, which Table is the following. Wherein

The firft Column contains the number of Points which are
fuppofed to have been thrown by A in three Raffles.

The fecond Column * contains the number of Chances
which A has to win if his Points be above jcjcjci, or the
number of Chances he has to lofe if they be either xxxi or
below it.

The third Column contains the number of Chances which
A has to lofe, if his Points be above xxxi, or to win if the/
be either xxxi or below it;

The Fourth Column contains the number of Chances
wliich he has for an equality ofX^hancc/

The Conftruftion of this Table cafily flows from the
confideration of the number of Chances which there are in a
fingle Raffle; whereof xviii or m, have i Chance ;xt'/i oviv^
3 Chances; xvi or i/, 6 Chances; xx/ or vi^ 4 Chances; xn;
or .v//| p Chances ; xlHov viii^ 9 Chances ; x/i or /x, 7 Chances ; .
xi or X, 9 Chances; which number of Chances being duly-
Combined will afford all the Chances of Three Raffles.

ATAH

the Doctrine 0/ Chancbi.

L B L E of all the CHANCES
'hich are in three Raffies.

Equality of
Chinee.

)r V

tx

i

X

xi

m m

Xl%

• • •

xnt
xiv

XV

xvi

xvii

xviii

XiX
XX

xxi

xxii

xxiii

xxiv

XXV

xxvi

xxvii

xxviii

xxtx

XXX

^ xxxi

Chances (DWia
or loie.

88475f
88471^

8S4681

884^34
884165

■ ' ^

883400
881954
879470

87ffOt
865)631

849705
834679

8ij39x

791506

^1 ■

761838
718971
690100

59947*

54886;
496314
441368

Cluncet to
'win or lofe.

15104

»3f37

35030

500J7
69344

^3130
II1898

155765
194636

137807
185164

388411

t

45

147
36^

18668

38871

43«7«
47457

Sum 441368
441368

884736

Tbii

. 111 1 M ^ii ij iM . i j iju j tj j i i u ii y i w "" i ^wqfP>>apiiw i w i j i" m. y w - jp^pyiwiw^iip^yf

»

\

The Doctrine 0/ Chances. Wi

This being once fuppofed, Ice ic be required to find the
Probability which A has of winning, when the number of
his Points being x/, there is but one Gamefter B befides
hiflifclf.

Take the number 791^05, which in the fccond Columa
(lands over againft the number x/, to be found in the Firft.
Take alfo one half of the number which in the Fourth Co-
lumn (lands over againd the faid Number xiy which half is
I ISA-)* ^^ ^^^^ ^^^o Numbers viz. 791506 and 1 1943 ^
added together, and their Sum 803449 being divided by
884736, which is the Number of all the Chances, the Quo-
tient, viz. lll^^l will exprefs the Probability required.

Now this Fradion being Subtra£ted from Unity, and the
remainder being ^VAV^ > ^< follows that the Numerators
of thefe two Fractions, a//>u 803449 and 81x87 do expreft
the Odds of winning, which may be reducM to 89 and 9
nearly.

But if the Number of Points which J has thrown for his
Chance being xl as above, there be two other Gamefters M
and C befides himfelf, the Probability which he has of win-
ning will be found thus.

Take the Square of the Number fet down over againft xl
in the fecond Column, which Square is 6x6481 74803 &
Take alfo the Produfl of that Number by the Number fet
down over againd xl in the Fourth Column, which Produft
is 1890591x216. Laftly, take the third part of the Square
of the Number fet down in the Third Column, which
third part will be i9oi8o}3x, and Ictall thofe numbers be
added together : Then their Sum being divided by the Square
of the whole Number of Chances, vijc by 78x757789696,
the Quotient ^^^^^^^^°^^^ will exprefs the Probability re-

quired ; from whence it may be concluded that the Oddsof
winning are nearly as 3 j to 7.

N. B. If fome of the laft figures in the Numbers of the
foregoing Table be neglefled, the Operation will be fliort-
ned, and a fufiicient Approximation obcainM by help of
rlis remaining Figures.

From what we have faid it follows, that A having xl foe
.the number of bis Points^ has lefs advantage when he Plays

T c againft

■■BP'wwwiiiT'iW'W'P'wrwtrarwwjw^i mvtmr^rmm^mff

1^2 7he DocTRiNK of Chances.

againft One than when he plays againft Two ; For fuppofing
each Man's Stake be a Guinea, he has in the firft Cafe 89
Chances for winning 1, and 9 Chances for lofing i :
From whence it follows that his Gain is -2^=^ or JL2.

which is about 1 7 A ^ ^'

But in the fecond Cafe, fuppofing alfo each Man*s Stake
to be a Guinea, he has 3 3 Chances for winning 2, and 7
Chances for lofing x:

Whence it appears, that his Gain in this Cafe is H^^

or ^ which is about i A — ixjb. — 8 rf. But Not^ that it

is not to be concluded from this fingle Inftance, that the
Gain of A wiU always, increafe with the number of
Gramefters,

If the number of Gameflers be never fo many, let f be
their nuqiber, let a be the number of Chances wnich <^ has
for winning when he has thrown his Chance, let m be the
number of Chances which there arQ for an Equality of Chance
between A and any of the other Gamefters ; Laftly, let the
whole number of Chances be denoted by S : Then the Pro-
bability which A has of winning will be expreffed by the
followmg Series.

which Series is compofed of the Terms of the Binomial
« + m^^ " * reduced into a Series, all its Terms being divided

^y ^f ^% h "h 5 ®^* refpeftively.

The foregoing Theorem may be ufeful, not only for fol-
ving any Cafe of the prefent Problem, but alfo an infinite
Varietv of other Cafes, in thofe Games wherein there \^
no Advantage in the order of Play : And the Application of
it to Numbers will be found eafy, to thofe who underftand
1k)\v to ;}fe LogarithpAS..

PRa

■t— '■ ■ >*<<* " * . ' *■ "» \n »» * tm''»f0mr''mm 9 ' 'm ^ _ immrm^m^ymnffitifV^^^^* ' ^''■*"* ' *^^^y^r^^ ^y . ^^ . y ■> m i'^*"'

■wwc

4«»»»-0«-«.

4

pe DocTRiMB qf Chancei; 'j^j^
PROBLEM L.

TO //I// tp^/f/ Prohahilitj there u^ tbdt Anj Numher of CerJs
of each Suit mxj be eontained in d given number ef tbtm
taken out of d given Stock

SOLUTiaN.

Flrfi^ Find the whole number of Chances there are for
taking the given number of Cards out of the givea
Stock.

Secondly^ Find all the particular Chances there are for
taking each given number of Cards of each Suit out of the
whole number of Cards belonging to that Suit.

Tbirdljj Multipl/ all thofe particular Chances together ;
then divide the ProduCl by the whole number of Chances^
and the Quotient will exprefs the Probability required.

Thus, If it be propofed to find the Probaoility of taking
Four Hearts^ Three Diamonds, Two Spades and Oat Club^
in Ten Cards taken out of a Stock containing Thirty-twa

Find the whole number of Chances for taking tea
Cards out of a Stock containing two and Thirty ; which is-
properly Combining two and Thirty Cards Ten and Ten^r
To do this, write down all the Numoers from \x inclufive*
ly to %x cxcluHvely, fo as to have as many Terms as there
are Cards to be Combined \ then write under each of them
refpeQively all the numbers from One to Ten inclufively,
thik^

1x2x3x4x5x6x7x8x9x10

Let all the numbers of the upper Row be Multiplied to-^
gethcr ; let alfo all the numbers of the lower Row be Mul-
tiplied together, then the firft Product being divided by
rhe fecondy the Quotient will exprefs the whole number of
Chances required, which will be Sa^ixx^q.

By the like Operation the number of Chances for ta-
king Four Hearts out of Eight, will be found to be

•»X »X3X4. - ' •

Xha

Wi^^TPMHi .H W. ■■

iW H B

* %P * ■ UL '1WW!ff*W!'gg*<****— ***— ^

ie?4 ^'^^ DocTKiNE of Chances.

The number of Chances for taking Three Diamonds out
of Eight will alfo be found to be ^^^^^ = 56.

The number of Chances for taking Two Spades out of
Eight will in the fame manner be found to be * ^^ 1= x8.

£4^^, The number of Chance/ for taking One Club out
of Eight will be found to be -^ =? 8.

Wherefore, Multiplying all thefe particular Chances to-
gether viz. 70, 56, 28, 8, the Produft will be 878080;
which being Divided by the whole number of Chances, the

Quotient /^~^i or —^ nearly, will exprefs the Probabili«

ty required : From whence it follows, that the Odds againft
taking Four Hearts, Three Diamonds, Two Spades and
One Club in Ten Cards, are vzry near 99 to a.

It is to be obferved that the Operatbns whereby the
Number of Chances is determined, may dways be contraft-
ed, except in the (Lngle cafe of taking one Card only of a
given Suit. Thus, If it were propofed to fliortea the Fra£tion

ia X ^i >>• 10 X 19 X aS X a? x itf x ay x 14 x n which de-
IX 2x3 X4X 5x6x7x8 X9K10'

termines the number of all the Chances belonging to the
foregoing Problem : Let it be confldered whether the Produft
of any two or more Terms of the Denominator, being
Multiplied together, be equal to any one of the Terms of
the Numerator ; if fo, all thoie Terms may be expunged
out of both Denominator and Numerator. Thus the Pro-
duQ of the three Numbers x, 3, ^ which are in the Deno-
minator, being ecjual to the Number X4, which is in the
the Numerator, it follows, that the three Numbers 2, j, 4
may be expunged out of the Denominator, and at the lame
time the Number 24 out of the Numerator. For the (ame
reafon the Numbers jr and 6 may be expunged out of the
Denominator, and the Number 30 out of the Numerator,
which will reduce the FraAion to be

31 X ^t X g'ly X 19 X a8 X 17 X a5 X af X 3g>if X a}
I X iL X 3 x/^xJx>Crx7XtX9 tTTo

It ought likewife to be considered whether there be any of
the remaining Numbers in the Denominator that Divide
exactly any of the remaining Numbers of the Numerator.

If

■•^*W^*«"'^~»i^^«<iWSi»<«««**n«>»«^»""fi*i»»-«^r^^^l«W«»*iw^^w»Hi^**i^^W

••wi^i**"

The Doctrine c/0{ancb€. 1^5

If (b^ thofe Numbers are to be expunged out of tfatf Deno-
minator and Numerator! but the refpeOive Quotients of
the Terms of the Numerator divided by thofe oTtbe Deno-
minator, are to be fubfticuted in the room of thofe Terms
of the Numerator. Thus the Terms 7, 8, and 9 of the De-
nominator dividing exa£Uy the Numbers aS^ix and zj of the
Numerator^ and the Quotients being 4, 4 zvA 3 reffyedively,
all the Numbers 7, 8» % xS, ax, %j oi^t to be expunged,
and the Quotients 4, ^ and } lubftituted in die room of xS,
3x1 X7 refpedivelyj in the following manoei)

4 4 S

xxzxjrx/fx^xiffx^xax^xto*

It ought alfo to be conddered, whether the remaining
Terms of the Denominator have anv comnaoa Divifor wi£
any of the remaining Terms of the Numerator ; if fp^ divid-
ing thofe Terms by their common Divifors, the relbeClive
Quotients ought to be fubftituted in the room of the Terms
of the Numerator, Thus, the only remaining Term in the
Denomiflator, befidesUbity^ being xo, which has a common
Divifor with one of the remaitibg Ternis of the Numerator,
oz/jc xf 9 and that common Divilor being 5, let 10 and xf
be refpeflively divided by the common Divifor c, and lee
the refpc£tive Quotients % and $ be fubftituted in the room of
them^ and the Fradion will be reduced to the foDowing, vU^

4 4 9 ^ f

Laftly, Let the remaining Number x in the Denominator
divide any of the Numbers of the Numerator whicH^ are
divifible by it^ fuch as x6, and let thofe two Numbers hh
expunged; bat let the Quotient of x6 by x, vix^ i}t be fubi.
ftituted in the room of x6: And then theFraCHoHi negleQ-
ing unity, which is the only Term remaining^ may be re*
duced to 4 X 31 X X9 X 4 x 3 X 13 x 5* x X3, the Produd:
of which Numbers is tf 45 1x240, as we have found itbeftxcL

The foregoing Solution being weD underftood, it will be
eafy to enlarge the Problem, and to find the Probability tii

U « taKiqg

'*»Wf— » i » i. i i ni.iwj *w'ipay*y^— ^»:«

•I H P H P . ■ l-l"!--

'j66 Ti&e Doctrine i>/ Chances. .

•

taking at Icaft Four Hearts, Three Diamonds, Two Spadcf,
and One Club| in Eleven Cards ; the finding of whicli de-
pends upon the four following Cafes, viz. taking*
y Hearts, 3 Diamonds, 2 Spades, i Club,
4 Hearts, 4 Diamonds, x Spades, j Club,
4 Hearts, 3 Diamonds, 3 Spades, % Club,
4 Hearts, 3 Diamonds, x Spades, 1 Clubs.
Now the number of Chances for the Firft Cafe will be
found to be 70x4641 for the Second 1097(00, for the Third
17^61^0, for the Fourth 3072x80-; which Chances being
added together, and their fum divided by the whole number
of Chances for taking Eleven Cards out of Thirty two,

the Quotient will be ,tS»448o " ^^^^ "^^^ ^ reduced to
•jj nearly.

From whence it may be concluded, that the Odds againft
the taking of Four Hearts, Three Diamonds, Two Spades,
and One Club, in Eleven Cards, that is, fo many at leaft
of every fort, is about px to f.

And by the fame Method it would be eafy to folve znyt
other Cafe of the like naturci let the mimber of Cards be
what it wiD;

PROBLEM LL

To pffd at PIQUET the FrctMity tpbht tbe Didler
has for taking Om Jea^ mr man in three Cardtp
hofvhfg^none in hie Hands.

SOLUTION.

FRom the number of all the Cards, which are Thirty
twoy fubt rafting Twelve which arc in the Dealere*
HairlSjthere remainsTwenty, among which aretheFourAces.
From whence it follows, that the number of all the
Chances for taking any three Cards in the Bottom, are
the number of Combinations which Twenty Cards may
afford, being taken Three and Three; which, by the Ruife^
given in the preceding Probleni| will be found to be

ax xx 1 ^* •*1^

The

(inpi> j i ^ Hii H I ] f / pt ■ ■ ■ii j *» i "ii<F'gii i i M ^|p t T^fW ^ iip M ii|W!Pn"!'^W>'tl^^i!WW*ip'^^^^f^'*^^

The Doctrine 0/ Chances. i6y

The number of all the Chances being thus obralned, find
the number of Chances for taking one Ace precifely, with
two other Cards ; Bnd next the number of Chances for taking
Two Aces precifely with any other Card ; Laftly, find the
number of Chances for taking Three Aces : Then thcfc
Chances being added together, and their fum divided by
the whole number of Chances, the Quotient will expreU
the Probability required.

But by the Diredions given in the preceding Problem^
it appears, that the number of Chances for taking One Ace

precifely are -7- or 4 ; and that the number of Chances for

taking any two other Cards arc '^^'^ or 1 xo : Froai

whence it follows, that the number of Chances for taking
One Ace precifely with any two other Cards, is equal to
4.x ixo or 480^-
la like manner it appears, that the number of Chances for

taking Two Aces precifely is equal to j W or 5, and

that the number of Chances for taking any. other Card is
il or 16; from whence it follows, tnat the number of

Chances for taking Two Aces precifely with any other Card
is 6 X x6 or 9^.
Laftly, It appears that the Number of Chances for taking

Three Aces is equal to -*iJ-5^ or 4.

Wheref6re tiio probability required will be found to be
, ^80 4- 9^4-^ or -i52-j which Fradion being fubtrafted
from Unity, tlic remainder, viz. -j^ will exprefe the Pro*

bability of not taking an Ace in Three Cards: From whence
it follows, that it is- 580 to 5^0, or 19 to x8, that the
Dealer takes One Ace or more in three Cards.

The preceding Solution may be very much contra£ied»
by inquiring at fird wliat the Probability is of not taking
an Ace in Tiirce Cards, which may be done thus:

The number of Cards in which the Four Aces arc con-
rained being Twenty, and confequently the number of Cards
out of which che Four Aces are excluded being Sixteen, it
follows, that the number of Chances which there are for the
taking Tbree^ Cards, among which no Ace {ball be found,

is-

"■^•'^•" . ■lJi(|»||i»»«tB^.iipB*ll*«f«PI*-«ilWf«iM^W*»Mi^"*«»!<"*"f*'

I^S the Do€TRINBi!)/CHANCB&

is the number of Combioations which Sixteen Cards may
afford, being taken Three and Three; which number of Com-
binations by the preceding Problem will be found to be

J^^'V^ '^ or k6o.

But the number of all the Chances which there are for
taking any Three Cards in Twenty, has been found to be
X X 40 i from whence it follows, that the Probability of not
taking an Ace in Three Cards is J^; and coniequently

that the Probability of taking One or more Aces in Three
Cards is -^^x The fame as bdorc.

IIAO

In the like manner, if we would find the Probability
which the Eldeft has of taking One Ace or more in his
Five Cards, he having none ia hit Hands; thtt feveraB
Chances may be calculated as follows*

Firft^ The number of Chances for taking One Ace aoA
Four other Cards will be found to be 7x80.

Seeondlj^ The number of Chances for taking Two Aces
and Three other Cards will be found to be 1360.

Thirdly^ The number of Chances for taking Three Aces
and two other Cards wiO be found to be 400.

Faurtbfyf The number of Chances for taking Four Aces
and any other Card will be found to be itf.

Laftlj^ The number of Chances for takuig any Five
Cards will be found to be 15504*

Let the fum of all the particular Chances^ vh^ ^^o
+ 3 )6o -f- 480 -4- 16 or 11136) be divided by the fum of
all the Chances, viz. by x 5504, and the Qpotient ii^ wiS

exprefs the Probability required.

Now the foregoing Fraaion being fubtraCted from Unity,
the remainder, viz. -^^ will exprefs the Probability of not

taking an Ace in Five Cards^ wherefore the Odds of taking
an Ace in Five Cards are 11136 to 43689 or 5 to x nearly.
But if the Probability of not taking an Ace in Five Ords
be at firft inquired into, the WotK will be very much

fliortcned; for it wiU be found to be 't k'iV^V^ x"
or 4)68, to be divided by the whofe number ot Chances,
viz. by X5504, which makes it as before^ equal to J^.

(

J

■V i' ' Vt>U il* vi *f*v ?•**" j»*— ''J^My< n i'»i " ^«w<«^';'^ . I iiiiiwiii . i»»*f , «— wnw"i»»i^^ppyw^pn^m^i<«w^t<q?'^^'^*T^

•f*««r

The DocTRiNB ^Cmajtcei ltf9

'But fuppofc ic were rccjuircd to find the ProbabiJitjr
which the Elded has of taking an Ace and a King in Five
Cards, he having none in his Hands/ Let the following
Chances be found^ riz.

1 jFor One Ace, One King and Three other Cards.

a For One Ace, Two Kings and Two other Cards.

3 For One Ace, Three Kings and any other Gird*

4 For One Ace and Four Kings.

y 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.

I X For taking any Five Cards in Twenty.

Among thefe Cafes, there being four P^trs that are al&e,
viz. the Second and Fifth, the Third and Eighth, the Fourth
and Tenth, the Seventh and Ninth ; it follows, that there are
only Seven Cafes to be Calculated, whereof the Firft, Sixth
and Eleventh, are to be uken fingly ; but the Second, Thirds
Fourth and Seventh^ to b^ douoled. Now the Operatioo
is as follows.

The Fir/J Cafe has ^ X
Chances..

fi y If X i»

Of 3fxo

The Secoffiy 4- ^ -f^ X'

I I X It
« X ft

or 1584, the double

1 X a

of which is 3168 Chances.

The Third, -i- x ^,^ll\ X -^ <>>^ ^9^ *« ^^^^
of which is )84 Chances.

The Fourth, -f. X ^:uiv

which is 8 Chances.

4<i ^ jlJLI- V -11

-^ or 4, the double of

The Sixths
The Seventh,

1 X s

I X ft ' I

4 X 1 X 1

1x2x3

- or 431 Chances;
or 14, the double of

which is 48 Chances.

The Eleventh, "r^r^.^x 'Ix 1 " ^^ « J5^4» being the

number of all the Chances for taking any Five Cards out

of Twenty.

X X From

I •,ii¥iitnrMr aii

r * I w tt "^ at P i l L. ■ I "

Tjo The Doctrine 0/ Chances.

From whence it follows, that the Probability whicfi the
Elded has for taking an Ace and a King in Five Cards,
he having none in his Hands^ will be expreft by the
Fra£tion

15504 15504

Let this Fraction be fubtra£led from Unity, and the re-
mainder being -^^t the Numerators of thcfe two Fraaions,

"vi^ 7.560 and 79449 will exprefsthe proportion of Proba«
bility that there is, of taking or not taking an Ace and a
King in Five Cards; which two numbers may be reduced
nearly to the proportion of ao to 3i.

By the fame Method of Procefs, any Cafe relating to
WHISK might be Calculated, tho* not To expeditioufljr.
as by the Method explained in the CoroBary of our XXtt
Problem : For which reafon the Reader is defired to have
recourfe to the Method therein explained, when any other
Cafe of the like nature happens to be propofod*

PROBLEM LIL

To find the Frohability cf taking mj number tf Sitits^ im
a given number cf Cards tdken out af d given Stoeki
without ffecifying what number of Csrds cf each Suit figB ii

taken*

SOLUTION;

SUppofe the number of Cards to be taken out of the given
Stock to be Eight, the number of Suits to be Four^ and
the number of Cards in the Stock to be Tbirty^twa

Let all the Variations that may happen, in taking One
Card at lead of each Suit, be written down in order, as
follow^

x> ^f t, Sf

h If »f 4>

^r »f 3> h

»» »f »> h

*> *i ?> »•

Tbeoi

»/*i;y»-^-^'^i.- v 1 ^ , i i iyp I M .:p j .y T w [ , -ra s ^- , - --j t ^B ^^^yw^iipi^iiga^piii^giy^i^ii^^

n)e Doctrine ^Chances. 171

Then fuppoHng any particular Suits to be appropriated
at pleafure to the Numbers belonging to the Firft Cafe, as
if it were required, for Inftance, to take One Heart, One
Diamond, One Spade and Five Clubs; let the Probability of
the fame be inquired into, which, by our Lsh Problem, wilt
be found to be .^11^21^; but the Problem not requiring the

Suits to be confined to any number of Cards of each Sorr^
it follows, that this Probability ought to be increafed in pro-
portion to the number of Permutations, or Changes of
Order, which Four Things may undergo, whereof Three arc
alike. Now thb number of Permutations is Four, and
confequently the Probability oftheFirfl Cafe, that is, of taking
Three Cards of three diiferent Suits^ and five Cards of a
Fourth Suit, in Eight Cards, will be the Fraftipn ^^^^^^ ■

multiplied by 4, or ,j;,V;;,.

In the fame manner the Probability of the Second' Cafe^

fuppofing it were confined to One Heart, One Diamond,

Two Spades, and Four Clubs, would be found to be J^i^A®. .
■ ' 10)18300'

which being multiplied by ix> viz. by the number of Pfer*
muutions which Four Sorts may undergo, whereof Two
are alike, and the other Two differing, it will follow, that
the Probability of the Second Cafe, taken without any re-^
ftriftion, will be expreffed by the Fraftion '^^\^^^\^*^
The Probability of the Third Cafe wiD likewife^bc found

to be ^12SL±2UL.

The Probability of the Fourth will be found to be

^'^Laflfy, The Probability of the Fifth will be found' to-
(jg .jiAliL^. Thefe Fractions being added together, their

i\xm^viz. ^/\yVo» will exprcfs the Probability of takings

the Four Suus m Eight Cards.

Let this laft Fraftion be fubtraded from Unity^ and the
remainder being ~^, it follows that 'tis the Odds of
76^3632 to 1864668, or 8 to 3 nearly, that the Four*
Suits may be taken in Eight Cards, out of a Stock con^
taining Thirty-tv/o.

The only difficulty remaining in thb matter, is thb fiodp*
ing readily the numoer of Fcrmutations^ which any. number *

of-

<

i " 1 . 1 I w m ■ i wn I ■■ '

«viPViaB*wac«««n««v<><'<^narnHniBi"Mi«imiii«M«iwMBM ^ ■ r*pMn«*t^n«Mpi^pmii

lyz /i&tf, Doctrine 0/ Chak<:bs.

of Things may undergo, when either they be all difTerenti,
or when fome of them be 'alike. The Solution of which
may be deduced from what we have faid in the CoroUarj
of our XVI I/A Problem, and may be explained as followsj
vn words at length.

Let all the numbers that are from Unity to that num«
bcr which cxprcfTes how many Things are to be Permu«
ted, be written down in order ; Multiply all thofe Numbers
together, and the Produ£l of them all will expre& the nuoi*
ber of their Permutations, if they be all different. Thus the
number of Permutations which Ten things are capable of, is the
Produ6l of all the Numbers iX2X3X4X5x6x7x>8^9Xi09
which is eaual to j6i88oa

But if (ome of them be alike, as fuppofe Four of One
fort, Three of another. Two of a Third, and One of a
Fourth, write down as before all the Nnmbers i x x x 3
X4X $'x6x7x8x9xio; then write under them as many
of thole Numbers as there are Things of the Firft fort that
are alike, which in this Cafe being Four, write the Numbers
1X1X2 X .4, beginning at Unity, and following in order.
Write alfo as many of thofe Numbers as there are Things
of the fecond fort that are alike, viz. 1x1x3, ftill be-
ginning at Unity. In the (ame manner write as many more
as there are Things of the Third fort that are alike, vizi
X X:^; and fo on: Which being reprefented by the Fra£^ion

jf X 1 X ^ X 4 X y X ^ X 7 X 8 X 9 X IP

iXaX3X4XxXaX3XxXaxt

let all the numbers of the upper Row be Multiplied toge-
ther, let alfo all the numbers of the lower Row be Multiplied
together, and the Firft Produft being divided by the Second,'
the Quotient 11600 will exprefi the number of Fbrmutations
required*

By this Method of Permutations, the Probability of
throwing any determinate number of Faces of the like fort^
witli any given number of Dice, may eafily be found. Thus,
fuppofe It were required to find the Probability of throwing
an Ace, a Two, a Three, a Four, a Five, and a She
with fix Dice. It is plain that there are as many Chances
for doing it, as there are Changes or Permutations in the

Order

7— .t i»i*«»wj««- «—-/»■ n«»«« ■••

ymm^ip^ "P I «•*.': '^ " ""'

The Doctrine 0/CHANCt& 175

Order or Place of fix difllcrent Things, fuppofe of the Six
Letters 4, b^ c^ df #, /, which by the Rule above given
would be 710, viz. the Product of the numbers !» X) 3^
^j jlf 6 : For tho' the Dice are not confidercd as changing
their Places, or as affording any Variation upon the toore
of the different Situation they may have in Refpefl to one an*
other, being thrown upon a Table ; yet they ought to be con-
fidered as changing their Faces^ which is equivalent to their
changing of Place. Now the number of all the Chances
upon Six Dice, being the number 6 Multiplied into it felf»
as many times wanting one as there are Dice, vizm
6x6x6x6^6x6 or 46656, it follows, that the Pro-
bability required will be expreft by the Fraftion lY'g$

and confequentIy» that the Odds againft throwing the Faces
undertaken, will be 46656 — 710 to 720, or 6410 i nearly.

In the fame manner fuppofe it were required to find the
Probabilit]^ of throwing One Ace, Two Two's and Three
Three's with Six Dice. The number of Chances for the doing
it being eaual to the number of Permutations whidi there
are in the fix Letters abicccy it follows, by the Rule before
delivered, that the number of thofe Chances will be 6O9.

t^/g, the Fraaion '^xlxlxlxUt > ^°* confequently
that the Probability required will be -j^i^ smd the Odds

againft the doing it 46^56 — 60 to 6O|0r 776 to i nearly.

If it were required to find the Probability of throwmg,
Two Aces, Two Two's and Two Three's with 6 Dice, the

number of Chances for doing it being ^xixlxlxlxf^
or 90, and the number of all the Chances upon Six Dice
being 46656, it follows, that the Probability required will
be expreft by the FrafHon y //^^.

Again* if it were required to find the Probability of
throwing Three Aces and Three Sixes, the number of

Chances for doing it being |xax|x?x Ixt ^^ *^ ^°*
the number of all the Chances 46656, the Probability re-
quired willt be. expreft^ by the Fra£lion -;^^^-

» •

Yy PRO-

ii

174 Ti^^ Doctrine (/Chancbs

PROBLEM LIII.

To fini at HAZARD the Chance of the Cafler^ n^»
the Mai/f being given^ be Throws to 4nj given number of
tolnts.

SOLUTION.

THis being eafily reduced to our XL VII/A Problem, k
is thought fumcient to exhibit the Solution of its dii«
fereot Cafes in the following Table, which (hews the Odds
for or againft the Caller.

MAlVf V.

exa£Uy nearlj^

AgainfttheCafter>^y28 t0 4O7 or yjXQxZ.
For the Cafter — 989 to 901 or 45" to 41.
For rhe Cafter — xipj to 1487 or 37 to 24*
For the Cafter — xips to 1487 or 37 to 24.
Againft the Cafter xz 1.7 to 1663 or 14 to ii.
Againft the Cafter X467 to 13 13 or 6x to 33;^

MAIN VL
Againft the Cafter 2879 to 1873 or 82 to 54:
Againft the Cafl:er 2483 to 2x^9 or 55 to 53.
For the Cafter —2621 to 2131 or 16 to 13.
For the Cafter — x6xi to X131 or 16 to 13.
Againft the Cafter X483 to XX09 or 5:8 to ^y *
Againftrthe Cafter 2483 to 2269 ,or 58 to 53.

MAIN VIL
Againft the Cafter 629 to 3^1 or 7 to 4^
Againft the Cafter 277 to xi8 or la to if.
For the Cafter -- 2f x to 244 or 36 to 35;
For the Cafter — 251 to 2i4'0r 3^ to 35^;
For the Cafter — 6or to 389 or 20 to 13^
For the Cafter ^ 263 to X3x or x/ to i;j

MAIN

•K^-^fflW*\t^

.,_..■ ■ •■-~r>- -«»>•>»

-••^^

^mgftfr*^^r*

■ W "iw ' ■»«■- ""*"

The DocTiiiNC«/ Chances. 17s

Poinlt

Thrmrn

to

i

Agaialt

n

Againft

Hi

For the

n/

For the

V

For the

vt.

For the

i

Agaioft

a

Aeamft

Hi

For the

For the

For the

vi.

Againft

mjiin vnt

thcCaftcr 317? to 1477 or
the Caller 148] to 1169 or
Caftcr — i6ii to 1131 or
Cafler — itfxi to »iji or
Cafter — 1483 to »itfg or
Cafter — »66y to 1087 or

MAIN IX.
tbe Cafter 3,467 to 1313 or
the Cafter s.117 to 1663 or
Cafter — *i93 to 1487 or
Caftcr — *x93 to 1487 or
Caftcr — 989 to 901 or
the Caftcr f Jo to 407 or

SI to aj.
j8 to jj.
t6 to 13.
16 to 13.
58 to 53.
83 to 6s*

6% to 33.

14 to II.

37 to 14.
37 to 14.

45 «> 4i*
37 to xS.

FINIS.

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