224? ESSAY ON PROBABILITIES. 



A dying before and after B, in any one given year, are 

 equal. We have therefore 



Or, 



But, (p. 223.) 



IAB...-JAB is AB|I 



Whence the final result is, 



RULE. For determining the value of 11. payable at 

 the end of the year of the death of A, provided B be 

 alive at the moment of A's death. 



(1.) From the value of a perpetuity subtract the 

 value of the joint annuity, and divide by that of a per- 

 petuity due. 



(2.) Multiply the value of the joint lives of B and a 

 year younger than A by the number alive in the table 

 at the younger age, and divide by the number alive at 

 the age of A. 



(3.) Repeat the preceding process, substituting B 

 for A, and A for B. 



(4.) From the sum of the results of (1) and (2) 

 subtract that of (3), and half the difference is the pre- 

 sent value required. The premium payable during the 

 joint lives is found by dividing the result by the value 

 of an annuity due on the joint lives. 



Though I have deduced the preceding rule at length 

 as an instance of the very improving exercise of deducing 

 the more complicated results of this subject by what 

 we may call the balance of annuities, yet for the rough 

 purposes of this work, if not for others still more exact, 

 the more simple process implied in finding A...B |AB 

 is sufficient. It must be obvious that the fraction of 

 the whole value of a survivorship insurance, which 

 depends on the risk of both parties dying in the same 

 year, is a small one ; so that it is nearly sufficient (and 

 certainly within the probable errors of tables, &c.) to 



