V. On the Calculation of Annuities, and on some Questions in the Theory of Chances. By J. W. LUBBOCK, Esq. B.A. [Read May 26, 1828.] THE object of the following investigation is to shew 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 individual 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 4689, (Price, Vol. I. p. 357.). The tables of Halley are founded upon the deaths which took place at Breslaw in Silesia during five years, and which amounted to 5869. 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 probability 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 preceding trials, the fewer the latter have been. Yo Laplace (Théor. Anal. des Probabilités, p. 426.) has investigated the method of determining the value of annuities, he there says, "Si l'on nomme y. le nombre des individus de l'age A dans la table de mortalité dont on fait usage et y le nombre des individus a l'age 4+x la probabilité de payer la rente à la fin de l'année A+X sera y" this hypothesis coincides with that I have before alluded to, as adopted by all other writers. Laplace, however, 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 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 shewing in 229250 deaths how each arose and another shewing 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 calculated tables of Annuities, he took the probabilities on which they depend, from a table of mortality founded on the deaths during five years at Breslaw. Since his time a great number of writers have treated of these subjects, of whom a notice may be seen in the Encyclopædia 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 decrements uniform; this is the case particularly with the Northampton 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 shew 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. 1 I annex four tables which I have calculated with the assistance of Mr. Deacon, from the tables of mortality for males and females at Chester, given by Dr. Price, Vol. II. p. 392. The two first tables shew 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 two last shew 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 shews that tables which would be applicable for the one would not be for the other. I have also subjoined a table comparing the values of annuities calculated from observations at Chester (according to the hypothesis 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 complicated. 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 balls, m, have been of the first colour, m, of the second colour, m of the third colour, m, of the pth colour. If x1, x2, x. . . . Xp are the simple probabilities of drawing in one trial, a ball of any given colour, the probability of the observed event, is The event being observed, the probability of this system of babilities is divided by the sum of all possible values of this quantity. pro The probability in n1+n....+n, subsequent trials of having n, balls of the first colour, n, of the second, n, of the pth, is a fraction of which the numerator is the sum of all the values of and of which the denominator is the sum of all the values of Since xx. . . . + x = 1, if x1, x2, &c. be all supposed to vary from 0 to 1, and all these values to be equally possible à priori, the numerator will be found by integrating the expression 2.... first from m (1 - X1 X2 X3... - xp-1)+", dx, x dx2....dxp-1 then from -2 = 0 to Xp-2 = 1-X1...... — Xp-3, and so on. The denominator will be found in the same way. If the coefficient of xxxma..... Xp P be called C, these integrations give for the probability required Cx .... .... .... (m1 + 1) (m,+2) (m, +3) ........ (m,+n,) (m2+1) (m, +2) ·(m2+n2).... (m1 +m2.... + p + n1 + n2 + n − 1) ' or if the product (mp + 1) (mp + 2).... (mp + np) be denoted by [m,+ 1]",, which is the notation used by Lacroix Traité du Calcul Differentiel, Vol. III. p. 121; the probability required is . |