LAPLACE. 603 



broken couples. The law of mortality is assumed to be the same 

 for men as for women ; and we suppose that the tables shew that 

 out of 7)1^ + 71^ persons aged A years, 77i^ were alive at the end of 

 T years, tti^ and 7ij^ being large. 



One mode of solving the proposed problem would be as follows. 



Take — as the chance that a specified individual will be alive 



771^ 4- Wj 



/ m \^ 

 at the end of T years ; then — will be the chance that a 



specified pair will be alive, and we shall denote this by 2?. There- 

 fore the chance that at the end of T years there will be v un- 

 broken couples, out of the original jii couples, is 



la 

 — — p^'Cl-pY-". 



fjb— V \v 



This is rioforous on the assumption that ^ — is exactly the 



^ ^ 711^ + 71^ -^ 



chance that a specified individual will be alive at the end of 

 T years : the assumption is analogous to what we have called an 

 inverse use of James Bernoulli's theorem ; see Art. 997. 



Or we may solve the problem according to the usual principles 

 of inverse probability as given by Bayes and Laplace. Let x 

 denote the chance, supposed unknown, that an individual aged 

 A years will be alive at the end of T years. We have the ob- 

 served event recorded in the tables of mortality, that out of m^-\-n^ 

 persons aged A years, m^ were alive at the end of J' years. Hence 

 the quantity denoted by y in Art. 1030 is 



m^ \7i^ 



and the quantity denoted by z is 



{xy{i-xy-^; 



therefore P = 



fjL— v\v 



, f V^ (1 - xY' (x'Y (1 - x'y-^ dx 



