208 



ESSAY ON PROBABILITIES. 



three lives, grant an additional annuity of II. on that 

 joint continuance. Thus, 



JA:B: C=|A + |B + |C-|AB-|BC-|CA + |ABC; 



in the process of finding which it becomes evident that 



ABC|A: B: C= | A + |B + |C- | AB- |BC- |CA 



It may he worth while to point out how, in every pos- 

 sible case, the preceding grants and withdrawals produce 

 the required effect ; namely, one annuity to be paid as 

 long as any one life lasts. The following are the pay- 

 ments and withdrawals : 



PROBLEM. Required C| A B, the value of an annuity 

 on the joint lives of A and B as long as they shall sur- 

 vive C. Grant a complete annuity on the joint lives, and 

 withdraw it while all three are alive. 



C|AB = |AB-|ABC 



PROBLEM. Required AB|C, the value of an annuity 

 on the life of C, to commence after the failure of the 

 joint existence of A and B. Grant an annuity on the 

 life of C, and withdraw it as long as all three are alive. 

 Thus, 



AB|C=|C-|ABC 



PROBLEM. Required the value of an annuity B : C|A, 

 to be paid to A after B and C are both dead. Grant 



