210 ESSAY ON PROBABILITIES. 



PROBLEM. ,An annuity on the longest survivor of 

 A and B, or | A : B, is to be equally divided between them 

 during their joint lives, and afterwards to go to the 

 survivor. What is the value of the interest of each ? 

 That of A is evidently \ (|AB) + B|A, which is 



J(|AB) + |A-|ABor|A-J(|AB) 

 Similarly that ofBis |B-J(|AB) 



which results are obtainable by a yet more evident 

 process, since the interest of each is evidently an annuity 

 on his own life, with half of it withdrawn as long as 

 both are alive. 



PROBLEM. To determine (BC)|A, the value of an 



annuity on the life of A, to commence with the failure 

 of the joint existence of B and C, provided it be B who 

 dies first. 



There are no tables for the accurate solution of this 

 problem ; but the following reasoning leads to a result 

 which cannot be far wrong, unless some of the lives be 

 very old, and which will always be near enough for the 

 species of application contemplated in this work. The 

 interest of A may be divided into two annuities, one 

 of which is B|AC, and the other a portion of B : C|A. 

 For A is certain of an annuity during C's life after the 

 death of B, and of another after both are dead, provided 

 B die first. Suppose A's interest in the latter annuity 

 to be worth one half of it, which is strictly true if B 

 and C be of the same age, and not much beside the 

 truth for considerable differences of age, particularly 

 when A is the oldest of the three. On this supposition 

 A's interest is 



B|AC+J(B: C|A) 



or |AC-|ACB+i(|A- |AB- |AC + |ABC) 



=(BC)|A 



which is obtained by supposing 



