STOP Early Journal Content on JSTOR, Free to Anyone in the World This article is one of nearly 500,000 scholarly works digitized and made freely available to everyone in the world by JSTOR. Known as the Early Journal Content, this set of works include research articles, news, letters, and other writings published in more than 200 of the oldest leading academic journals. The works date from the mid-seventeenth to the early twentieth centuries. We encourage people to read and share the Early Journal Content openly and to tell others that this resource exists. People may post this content online or redistribute in any way for non-commercial purposes. Read more about Early Journal Content at http://about.jstor.org/participate-jstor/individuals/early- journal-content . JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not-for-profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact support@jstor.org. 1855.] 197 On the Calculation of Annuities, and on some Questions in the Theory of Chances. By J. W. Lubbock, Esq., B.A.* [Extracted, by permission of the Author, from the Transactions of the Cambridge Philosophical Society.] 1. ±HE object of the following investigation is to show how the probabilities of an individual living any given number of years are to be deduced from any table of mortality. All writers (with the exception of Laplace) have considered the probability of an indi- vidual dying at any age to be the number of deaths at that age recorded in the table, divided by the sum of the deaths recorded at all ages. This would be the case if the observations on which the table is founded were infinite; but the supposition differs the more widely from the truth the less extended are the observations, and cannot, I think, be admitted where the recorded deaths do not altogether exceed a few thousand, as is the case in the tables used in England. The number of deaths on which the Northampton Tables are founded is 4,689 (Price, vol. i. p. 357). The tables of Halley are founded upon the dea'ths which took place at Breslau, in Silesia, during five years, and which amounted to 5,869. If a bag contain an infinite number of balls of different colours in unknown proportions, a few trials or drawings will not indicate the proportion in which they exist in the bag, or the simple pro- bability of drawing a ball of any given colour; and not only the probability of drawing a ball of any given colour, calculated from a few observations, will be little to be depended on, but it will also differ the more from the ratio of the number of times a ball of the given colour has been drawn, divided by the number of the pre- ceding trials, the fewer the latter have been. Laplace (Theor. Anal, des Probabilites, p. 426) has investi- gated the method of determining the value of annuities. He there says — " Si l'on nomme y„ le nombre des individus de Page A dans la table de mortalite dont on fait usage, et y x le nombre des indi- vidus k l'age A + x, la probabilite de payer la rente a la fin de l'annee A + x sera — ." This hypothesis coincides with that I have y<> before alluded to, as adopted by all other writers. Laplace, how- ever, means this as an approximation, for he has investigated differently the probability of an individual of the age A living to the age A + a (p. 385 of the same work). He there considers two cases only possible ; but as an individual may die at any instant * Now Sir John William Lubbock, Bart., F.R.S., &c. 198 On the Calculation of Annuities, [April during life, I think it may be doubted whether this hypothesis of possibility should be adopted. Captain John Graunt was the first, if I am not mistaken, who directed attention to questions connected with the duration of life. He published a book in 1661, entitled Observations on the Bills of Mortality, which contains many interesting details, although it is written in the quaint style which prevailed in those times. In this book, amongst other tables there is one showing in 229,250 deaths how each arose ; and another showing of 100 births " how many die within six years, how many the next decad, and so for every decad till 76" — which is in fact a table of mortality, and is probably the first ever published. After Captain Graunt, Sir W. Petty published his Essays on Political Arithmetick. Halley, however, was the first who calcu- lated tables of annuities : he took the probabilities on which they depend, from a table of mortality founded on the deaths during five years at Breslau. Since his time a great number of writers have treated of these subjects, of whom a notice may be seen in the Encyclopaedia Britannica, or in the Report from the Committee on the Laws respecting Friendly Societies, 1827 (p. 94). It is to be regretted that those who have published tables of mortality should generally not only have altered the radix or number of deaths upon which the table is constructed, but also the number of deaths recorded at different ages, in order to render the decre- ments uniform; this is the case particularly with the Northamp- ton Tables, as published by Dr. Price (see Price on Reversionary Payments, vol. i. p. 358). For if observations were continued to a sufficient extent, they would probably show that some ages are more exposed to disease than others — that is, they would indicate the existence of climacterics, of which alterations such as these destroy all trace. I annex four tables,* which I have calculated, with the assist- ance of Mr. Deacon, from the Tables of Mortality for Males and Females at Chester, given by Dr. Price (vol. ii. p. 392). The first two tables show the probability of an individual at any age living any given number of years, as well as the expectation of life at any age. The last two show the value of £1 to be received by an individual of any age after any number of years, and the value of an annuity. The difference between these values for a male and female, is very great, and shows that tables which would be appli- cable for the one would not be for the other. * See page 207, and the note there. 1855.] and on some Questions in the Theory of Chances. 199 I have also subjoined a table comparing the values of annuities calculated from observations at Chester (according to the hypo- thesis of probability I have assumed), with some which have been calculated from observations at other places. Until lately, the Government of this country granted annuities, the price of which depended on the price of stock, which renders their tables com- plicated. I have given their values of a deferred annuity for five years, compared with those I have calculated from the observations at Chester : it will be seen that the former are much too high. 2. Suppose a bag to contain a number of balls of p different colours, and that, having drawn m x -\-m 2 + m 3 + m p balls, m x have been of the first colour, m 2 of the second colour, m 3 of the third colour, m p of the p th colour. If oc X) x 2 , x 3 . . . x p are the simple probabilities of drawing in one trial a ball of any given colour, the probability of the observed event is x™y x x 2 m > • • • • x x p m p, multiplied by the coefficient of Xi m tx 2 m i x p m r in the development of {x x + x 2 . . . + x p ) m i +m z- •■■+ m P . The event being observed, the probability of this system of probabilities is x^i x ar 2 m * • • • • x x t m p, divided by the sum of all possible values of this quantity. The probability in n x -\-n 2 . . .. + n p subsequent trials of having w, balls of the first colour, n? of the second, n p of the p & , is a fraction of which the numerator is the sum of all the values of ^.^,+w, x# 2 "» 4 +» s . . . .xx p m p +n p, and of which the denominator is the sum of all the values of j," 1 ! xx 2 m * .... xx p m p, multi- plied by the coefficient of ^"i x #2% .... xx p n p in the develop- ment of (Xi+X 2 + X 3 ....+X p ) n l +n *~~ +n P. Since x x + x 2 . . . .+x p =l, if x u x 2 , &c. be all supposed to vary from to 1, and all these values to be equally possible a priori, the numerator will be found by integrating the expression Xl m \+"\ x a^™ 2 " 1 "" 2 - • •(! — *i — x 2— «3 — x r -\T r + " p d*\ xdx 2 ...x dx p _ x first from x p _ 1 =0 to x p ^ 1 = l— x x — x 2 — x p _ 2 , then from x p - 2 =0 to :rp_2=l — x y — Xp-z, and so on. The deno- minator will be found in the same way. If the coefficient of x x m \ x x 2 m z . . . x x p m p in the development of (Xi + x 2 . . . . +x p ) n i +n f"+ n p be called C, these integrations give for the probability required (m i + l)(m 1 + 2)(m l + 3)...(m 1 +n i )(m 2 + l)(m i + 2)...(m 2 +n 2 )... Cx (mi + mi+nii + m p +p)(m 1 + m 2 +m 3 + m p +p+l) (m p +l)(m p +2) (m p +n p ) (mi+m 2 +p+n l + n 2 +n 3 — 1)' 200 On the Calculation of Annuities, [April or if the product (m p +l) (m p + 2) .... (m p + n p ) be denoted by [m p +l]"p, which is the notation used by Lacroix (Traite du Calcul Differentiel, vol. iii. p. 121), the probability required is c . Qi + lM^ + l]" 2 - •••[>,+ 1]V [j», + W 2 m p +^]"l+"2+"3 • • •+", This probability is the same as if the simple probability of drawing a ball of the p & colour were m p + 1, with the difference of notation. When n 2 , n 3) « p _i, &c. = 0, and n p = 1, this expression gives for the chance of drawing a ball of the p 01 colour m p + l mi + m 2 .... + m p +p' and the probability that the index of the colour drawn is be- tween n— 1 and n + q + 1 is m n +m n+l .... m n+t +q mi + mz .... +m p +p If we suppose the law of the possibility of life to be such that p cases or ages are possible — a priori, m lt m n , &c. will be the number of recorded deaths in a table of mortality at those respective ages, and the chance of an individual living beyond the » th age will be »>»+>»„+i .... m p +p— n mi + nti .... +m p +p m n + m n _i + &c. + m p is the number given by the table as living at the n th year ; therefore, on the hypothesis of this law of possibility, the chance of an individual living beyond the w th year is a fraction of which the numerator is the number living at that age, +p—n, and the denominator is the whole population on which the table is founded, or the radix + p. The Tables I. and II. have been calculated from this formula, from observations at Chester given by Dr. Price (vol. ii. p. 107) : p was taken equal to 101 for a child at birth — that is, the chances of a child living beyond a hundred years, and of its dying in each intermediate year, were supposed to vary from to 1, all these values being equally probable, a priori. The value of any sum to be received after any number of years is equal to the sum itself, multiplied by the chance of the individual being alive to receive it : therefore these tables give the value of unity to be received after any number of years. Considering duration of life to be valuable in proportion to its Cx 1855.] and on some Questions in the Theory of Chances. 201 length, the value of the expectation of life to any individual is the sum of the chances of his living any number of years multiplied by the intervening time ; so that if P„. be the chance of an indivi- dual living exactly n years, the value of his expectation of life is 2nP„, which is evidently equal to SP'„, if P'„ be the chance of an individual surviving n years : therefore the value of the expecta- tion of life of any individual is the sum of the numbers on the same line in Tables I. and II. The unity of expectation is here the expectation of an individual who is certain to live exactly one year. The Tables I. and II. give the values of contingencies depending on a single life, without discount ; the Tables III. and IV. are the same values, discounted at the rate of three per cent. compound interest. These tables give the values of annuities about six per cent, higher than those calculated from the North- ampton, and given by Dr. Price, vol. ii. p. 54. The only tables of annuities on female lives that I have met with are calculated from observations in Sweden, and are given by Dr. Price, vol. ii. p. 422 ; but they are calculated at four and five per cent, interest. It is not to be expected, however, that tables calculated from observations made in one country will serve in another, or even in different parts of the same country.* The probability of having ra, balls of the first colour in % + N trials, the colours of the other N balls being any whatever, is fxf^jl-x^fx^xfs (l— Xl — X2 x^-^pdx^x^ dx p _ x fxi'ix^" 2 . . . .(1 — Xy — x 2 . . . .x p _ l )'"pdx l dx 2 . . . .dx p _i ' multiplied by the coefficient of x n \ in the development of { x i+y) n ^> the integrals being taken between the same limits as before. These integrations give for the probability required (»»i + l)(>»i + 2). ..(wi+7tiX»»2+'»3+'»4. ..+p— l)(m 2 +m 3 +m 4 . ••+/>). {mi+m 2 + m p +p)(m l + m 2 +m p +p+l) (m 2 +m 3 + m 4 +jp+N— 2) (j»,+m 2 + m 3 +p + m + N— 1)' C being equal to k+l> ^V///.^ ' Ado P tin S the same notation as before, this probability is equal to * Since writing the above, I find that Mr. Finlaison has given the values of annuities, distinguishing the 6exes, in the Report of the Committee on Friendly Societies, 1825, p. 140. VOL. V. P 202 On the Calculation of Annuities, [April [m l +iyi[m i +m 3 ..+m p +p— l] N [»ii + »n 2 + TOa «i p +jh]"i +n CQi + lJ"^-^ +m p +p— l] CT i +1 ~~ [m 2 +OT 3 ....+N+i»-l]"i + '" 1+1 which probability, as before, is the same as if the simple probability of drawing a ball of the />*** colour were »ip + l. If m 2 + m 3 . . . . + m p +p—2=M., and if n, and N are in the same ratio as »i, and M, the chance that the number of balls of the first colour in «i + N trials is between the limits n x and n x ±z, by the reductions given in the Thiorie Anal, des Probabilites, p. 386, is 1—21/ — — i—- +f — - / dze 2m 1 M(N+» 1 )(M + N+m 1 +n 1 ), e being the number of which the hyperbolic logarithm is unity, and the integral being taken from z=z, to ^=infinity. The question of determining the probability that the losses and gains of an Insurance Company on any class of life are contained within certain limits, is precisely similar to this. It will be seen from the formula "»*+ "Wi- •••"»»+? + g (p . 2 00, mt + nit + m p +p xr line 12), that if life were divided into an infinite number of ages or intervals (in which case p is infinite), the hypothesis of possibility remaining the same, the probability of an individual dying in any given interval would be the given interval divided by the whole duration of life, which coincides with that which is given by De Moivre's hypothesis. Thus if life were supposed to extend to a hundred years, the probability of an individual dying in any given year would be -j-^-g, and any finite number of observa- tions or recorded deaths would not influence the value of this probability. As diseases and other causes producing death are not equally distributed throughout life, the last hypothesis cannot be adopted. In order to investigate accurately the probability of death at any age, it would be necessary to know the law of possibility. Let <j>pX p be the probability of the possibility of x v : then the pro- bability in the former question of having n x balls of the first colour, w 2 of the second, &c, in n 1 + w 2 • • • • + w j> trials, is yy l+ "'(frgi>2 » + "Kf i ig!i)...(l— x l -x i ...x„_i)'"'+ n pdx i dx 2 — dx p _ x /*i"i(0i*i), «™s(^ 2 iF 2 ) (1 —xi —x 2 . . .x p ^) m p dxidxi —dx p ^ ' 1855.] and on some Questions in the Theory of Chances. 203 is a sign of function, and this function may be either continuous or discontinuous. This expression must be integrated between the same limits as before. The coefficients of the different powers of x p in <j>pX p , or the constants in <j>pX p , will generally be functions of the index p. If the probability of life were known at a great many places, and if x Pl were the value of x p at q v places, x P9 at q t places, &c, the law of possibility might be determined approximately by considering QpXp as a parabolic curve, of which x p is the abscissa, passing through the points, of which the ordinates are ?1 ?2 ?1 + ?2 +> &c- ' 7i + ft +> &c. ' 3. In the preceding investigations, the results of the preceding trials are supposed to be known ; it may be worth while to examine what the probability of any future event is when the results of the preceding trials are uncertain. Let a bag contain any number of balls of two colours, white and black: suppose m trials have taken place; and let e„ be the pro- bability that a white ball was drawn the /1 th trial, /„ the probability that a black ball was drawn. e„+f n =l. First let e lt e 2 . . . . e„ be all equal, and let x be the probability of drawing a white ball. If a white ball was drawn every time in the m trials which have taken place, the probability in «i + n 2 future trials of having n x white balls and » 2 black balls is («i+«2)(wi+n 2 -l)---("i + l)/^- | "''(l-^" t ^ 1.2 n n faf'dx But the probability that a white ball was drawn every time is e m ; therefore the probability of drawing a white ball n t times and a black ball n 2 times, on this hypothesis, multiplied by the pro- bability of the hypothesis, is (nt+nzXnx + nz— l)...(ni + l) „ /^ m+ °» (1— x) n 2dx 1.2 n 2 e ~JaFTx ; and the probability of drawing n x white balls and w 2 black balls will be the sum of the probabilities on every hypothesis, multi- plied respectively by the probability of the hypothesis, which is p2 204 On the Calculation of Annuities, [April (wi + H^fa + n,-!).. .(«i + l) ( m f<r**jl-xpd* 1.2 « 2 \ e /V<fo «/ /ar-\l—x)tb "*" 1.2 ^ faf-Xl-&yki ' This integral being taken from a;=0 to 03=1, is (» 1 +«g)(wi+n a — l)..(wi + l) r w 2 ,n 2 — 1,^— 2 l.m+1 1.2 n 2 \m+n 1 + l.m+n 1 + 2...m+n 1 + ii2+l ni+\.iu.iu — 1....2 +me m -]f. — 2.J! m+l.m+, &c. J» + »!.»» + Wi + l . . . .Wi+»]+W 2 _ (wi + w 2 )(wi+w 2 — 1) (wi + 1) 1 1.2. .. . n 2 m+2.w+3. . .w + » 1 + «2+i |n 2 .»! 2 — 1 .n 2 — 2. ..m+ni.m+ni — 1. ..m+1 .e m +n 2 -i-l ■ •• 2.m+n x — 1. . .m+1 .m .me m ~]f+ , &c. _,,. . . lx rf n . + " 2 .« B 3 a^i(ex+/w)"' , , This series is equal to , - , . ±3d— when x and ^ dx n \dy n i y are made equal to 1, and this is equal to 1 . 2 . 3 . . . . n x . 1 . 2 . 3 . . . . ra 2 x coefficient of A"i k n *, in the development of (i+ hp(i + A)xi + eh +/ty (l +e h+/ky=l+m(eh+fk)+^^-(eh+/ky m.m — l.m — 2, , „,,„ + j-^3 («*+/*)»+ , &c (1 + hfi(l+kp=h"iJ^+nih' , i- l k"2+ Wl '" 1 ~ 1 A"i-**-» + n 2 &"i/E"2- ■ + » . w^".-'^- 1 + - '"' ' "' ~ * A-i- 2 ^-' + , &c. Coefficient of h" i A"* = 1 + Jn(n,e + n 2 /) m.m—l( w,.^— 1 n 2 .n 2 —l\ 1855.] and on some Questions in the Theory of Chances. 205 The probability required is 1.2.3 «i+«2 m+2.m+3. . . .m+ni + t^+l l+ m (n l e+T h f)+ — | - l - e i +2n l n 2 e/+ 2 - » g f A+, &c. If there are p different colour's, and if m trials have taken place, and e g , p is the chance that a ball of the p th colour was drawn the q th trial, the probability of drawing n x balls of the first colour, n 2 of the second, n P of the p^, in n t + n 2 . . . . +n p future trials, may be found in the same way. Let «1,1 +«1,* + «1,I+ &c «l,n = S 1 , «l, e l, 1 J *1, 2 "I" *1, 3 » e l, 4 < ^ C> = "2> *1> (the sum of the products of e 1 two and two together,) «1,1> «2,S + *1,8> «2,3+»&C. =S 1 « 1 , SAj* and so onj then it may be shown that this probability is equal to 1.2.3.n 1+ n 2+ n 3 . ■ + n, (1+Sei) „ l(1 + S ^.. .. (1 + S ^, >n+^ w+w 1 +n 2 .. ..+n p +p— 1 1 + (Sej), 1 + (Se 2 ), &c. being expanded by the binomial theorem, and the indices of S written at the foot. The method which was used for summing the series in the last page is of very general application, and depends, in fact, on this principle, that the generating function of the sum of any series is» the sum of the generating functions of each of the terms of the series. If in the last formula « 2 , n 3) &c.=0, and if there be only two events possible, and n x = 1, the probability required is — — ■— . In order to apply this, suppose an individual to have asserted m events to have taken place, of which the simple probabilities are equal, and equal to p ; and suppose it required to find the proba- bility of his telling the truth in another case, where the simple probability of the event he asserts to have taken place is not known. Let x be the veracity of the individual, the probability of his telling the truth on this hypothesis is — ,., — . ,., r : and ° Jr px+(l—x)(l—p) the probability of his telling the truth is the sum of the pro- babilities of his telling the truth on each hypothesis, divided by the number of the hypotheses. * This is a method of notation which ohtains, but it is not meant to imply that S 1 e 1 S 1 e 2 =S,,e 1 x S„ e 2 . 206 On the Calculation of Annuities, [April Suppose x to vary from to 1, and all these values of x to be equally probable a priori, then the probability of his having told the truth and the event having taken place is / — i-r. — ttt r, r J px+(l'—p)(l—x) taken from x=0 to x=l, which integral is cM'-s^Ty- 9 If p = -^, this probability is '81601. Generally, Up > £, the assertion that the event has taken place (on this hypothesis of veracity) rather diminishes the probability that the event has taken place; if p= J, the assertion does not alter the probability; if p < \, the assertion rather increases it. 9-1601 9 1 + Se If P=^> «= -81601, let jh=10; then tlf^ r 10 m+x 12 which is the probability that the individual will tell the truth in another case. If the individual had told ten truths, the chance of his telling the truth in another case would have been — • All values of x between and 1 were supposed equally pos- sible : if they are not, let <j>x be the probability of the possibility of any value of x ; then the probability of an individual telling the ffxdx truth will be 'px+(l — x) (1— p) grals being taken from x=0 to <e=1. divided by f$xdx, these inte- Table formed from the Burials in All Saints' Parish, Northampton, from 1735 to 1780. (Seepage 198, line 26.) Age. Actual Number of Burials. Reduced to radix 1 1050. By the Observations. As altered by Dr. Price. Under . . 2 Between 2 and 5 S „ 10 „ 10 „ 20 „ 20 „ 30 30 „ 40 „ 40 „ SO . ,, SO „ 60 60 „ 70 • „ 70 „ 80 „ 80 „ 90 „ 90 „ 100 Total . . . 1,529 362 201 189 373 329 365 384 378 358 199 22 37981 899J 499* 469| 926| 817| 906} 954 939J 889| 494| Mi- 4367 1034 574 543 747 750 778 819 806 763 423 46 4,689 ll 650 11650 1855.] and on some Questions in the Theory of Chances. 207 Table I. — Males. Table II. — Females. Expectation of Life, by the Author's method. Expectation Age. of Life, by usual method. 29-75345 2813 1 36-69541) 35-76 2 40-21306 39-42 3 42-54615 41-97 4 43-83928 43-33 5 44-09357 43-20 10 42-75204 41-92 15 38-95786 38-05 20 35-82561 34-86 25 32-99367 32-00 30 30-27118 29-25 35 27-04332 25-97 40 2404172 22-92 45 21-34876 20-20 50 18-81444 17-64 55 16-34857 15-14 60 13-63392 12-36 65 12-05917 10-79 70 9-41263 8-05 75 8-43636 7-00 80 6-99009 5-43 85 5-90384 4-25 90 4-32000 2-50 95 2-14285 1-00 Table III.— -Males. Age. Value of Annuity by the Author's method. Value of Annuity by usual method. 13-96256 1 17-35468 2 19-17322 3 20-38907 4 21-15972 5 21-42118 21-283 10 21-55443 21-512 15 2038198 20-283 20 19-42818 19285 25 18-55566 18-399 30 17-67138 17-492 35 16-36473 16134 40 15-05567 14-667 45 13-81590 13-493 50 12-59164 12-316 55 11-28375 10-866 60 9-63491 9-140 65 8-77709 8-220 70 6-94786 6-260 75 6-17140 5-291 80 5-11141 85 4-30505 Expectation Expectation * Age. by the Author's method. of Life, by usual method. 34-55535 33-27 1 40-04475 39-54 2 43-65276 43-25 3 45-87700 45-68 4 47-23533 4711 5 47-99860 47-44 10 45-69310 45-17 15 41-96030 41-36 20 38-76308 38-10 25 35-49329 34-78 30 32-79.565 32-27 35 3000384 29-26 40 27-13292 26-37 45 24-29072 23-50 50 21-43212 20-62 55 18-35900 17-52 60 15-09954 14-20 65 12-83834 11-94 70 9-78378 8-81 75 8-12794 7-14 80 6-30434 5-20 85 5-97402 4-81 90 4-55263 3-46 95 2-07692 1-71 Table IV. — Females. Age. Value of Annuity by the Author's method. Value of Annuity by usual method. 15-75290 1 18-42550 2 20-14850 3 21-32367 4 22-11858 5 22-64306 22-624 10 22-41169 22-439 15 21-25267 21-235 20 20-36435 20-323 25 19-33571 19-265 30 18-65687 18-583 35 17-62920 17534 40 16-56779 16-366 45 15-42043 15-282 50 14-14872 13-983 55 12-58232 12-376 60 1067266 10-410 65 9-37655 9080 70 7-26287 6-889 75 6-02689 5-338 80 4-65536 85 4-34792 Note. — We do not reprint all the tables originally given in this paper. They are of value now only as exhibiting the results of the methods laid down by the Author, and for that purpose those here quoted will suffice. — Ed. A. M.