222 ESSAY ON PROBABILITIES. 



at the beginning of the year. This may he denoted 

 by | (A...B). 



The condition that a life shall be alive at the begin- 

 ning of the year must be, in tables of averages,, the 

 same as that of a life a year younger being alive at the 

 end of the year. For example, suppose that of 500 

 persons of the age of n I, 493 attain to that of n, 

 then 500 annuities granted on the lives of A n -\ will 

 be equivalent to 493 granted to A w , if the latter be 

 payable at the end of any year in which A shall have 

 been alive at the beginning ; and the same for any 

 joint lives combined with both. Thus 500 annuities 

 granted on the joint lives of A n _i and B, payable as 

 usual, are equivalent to 493 on the joint lives of A n 

 and B, payable as long as B is alive at the end, and A 

 was alive at the beginning, of a year. Hence | (A...B) is 

 .5.00. o f |A,B, where A / stands for a life a year younger 

 than A. Hence the following RULE. To solve the 

 preceding question, multiply the value of a joint an- 

 nuity on B and one year younger than A by the number 

 alive at that younger age in the table, and divide by 

 the number alive at A's age : the result is the present 

 value required. Or more concisely, divide | (A X B) by 

 the chance which A, has of living a year. 



Now let us ask, by how much does |(A...B) as above 

 described exceed | AB. The only possible case in which 

 a payment will ever be made upon the first annuity, and 

 not upon the second, is when A dies before B, for both 

 are determined by the death of B. When A dies be- 

 fore B, the annuity will be paid at the end of the year 

 upon |(A....B), if B be alive, but not upon | AB. Con- 

 sequently the excess of the first over the second is the 

 value of l to be received at the end of the year in 

 which A dies, provided B be then alive. 



r '_ 



A|TB=|(A...B)-|AB, B|lA=|(B...A)-|AB 



Also the present value of an annuity to be paid at the 

 end of any year in which both A and B were alive at the 



