190 ESSAY ON PROBABILITIES. 



class of which just one half die in ten years. A 

 repetition of a similar process for every year in which 

 the annuity may become payable gives the present 

 values of the other payments. The sum of all the 

 present values of the different payments is the present 

 value of the whole annuity. Thus if we take a very 

 old life, and suppose that of 10 alive at the present 

 time there will be left at the end of successive years, 

 7, 5, 3, 0, there are three possible payments of the 

 annuity, and the chances of having to make them are 

 j-?,-, -, and -p () . But the first, if made at all, is not 

 made for a year, and the second and third are not made 

 for two and three years. If then a, b, and c be the 

 present values of II. to be received at the end of one, 

 two, and three years, the values of the several payments 

 are 1 7 fT of a, T 5 - of &, and -^ of c, the sum of which is 

 the value of an annuity of 11. on any one life of the kind 

 in question. 



By the status of an annuity, I mean the state or 

 condition of things during the continuance of which 

 the annuity is to be paid. This status may be simple 

 or complicated : in the latter case the method of find- 

 ing the chances of its continuance or termination will 

 also be complicated ; but this does not affect the simple 

 rule by which those chances, when found, are made, in 

 conjunction with the rules of compound interest, to give 

 the value of the annuity. Thus the status in question 

 may be of a compound character, allowing of several 

 distinct changes. For instance, A is to enjoy an 

 annuity to the end of his life, unless B should die 

 before C, in which case it is to cease. This annuity will 

 be enjoyed as long as either of the following status exist. 



A, B, and C all living. 



A and B living, and C dead. 



A living, and B and C dead, C having died first. 



Tables of the values of annuities are constructed in 

 the case of single lives and two joint lives; that is to 

 say, the mere inspection of the tables will enable us to 



