ON A QUESTION IN THE THEORY OF 

 PROBABILITIES*. 



THE following question affords a good illustration of the 

 methods employed in the more difficult parts of the theory of 

 probabilities. In a paper presented to the Philosophical Society, 

 I applied the kind of analysis we are about to make use of, to 

 the celebrated Rule of Least Squares. There is, in fact, a close 

 analogy between the two investigations. Laplace's solution of 

 the present question is obtained by a process similar to that 

 which he had employed when treating of the best method of 

 combining discordant observations. 



What is the probability that the sum of the times which 

 each of n persons has respectively yet to live will amount to 

 a given time T? 



Let $ p x p dx p be the probability that thep th person will live 

 precisely a time x p longer, (f> p denoting some function of x p , 

 which is necessarily such that 



<l> p x p dx p = l, 

 as it is certain that he will die at some time or other. 



Let x lt a? 2 , ... x n be so related that 



The probability of this particular combination is 

 <fri x i <t>A faXndx^ dx 2 ... dx n , 

 or <Mi<M 2 *. ( T -XI - aV-i) dx i " dx > 



and the aggregate probability sought is the integral of this 

 expression obtained by giving all possible positive values to 



* Cambridge Mathematical Journal, No. XXI. Vol. iv. p. 177. May, 1844. 



