A N X 



2. Of Life Annuities. (Wood.) 



\. To find the probability that an individual of a given age will live 

 nny number of years. 



Let A be the number in the tables of the given age, B, C, D ......... X 



the number left at 1, 2, 3 ......... t years ; then is the probability that 



p 



the individual will live one year ; -j- the probability that he will live 



two years, ~ that he will live t years. Also ~~-, T~ ~~T~ 



A A A A 



are the probabilities that he will die in 1, 2, t years. 



2. To find the probability that two individuals P and Q, whose ages are 

 known, will live a year. 



Let the probability that P will live a year, determined by the last Art. 

 be , and the probability that Q will live a year ; then the probabi- 



lity that they will both be alive at the end of that time is - . 



mn 



3. To find the probability that one of them at least will be alive at the 

 end of any number of years. 



Let be the probability that P will live t years, and the probabi- 

 lity that Q will live the same time ; then the prob. that one of them at least 



will be alive at the end of the time is 1 P "" L 9 ~ 1 , or jP + g '~ I - 



pq pq 



4. To find the present value of an annuity of 1. to be continued dur- 

 ing the life of an individual of a given age, allowing compound interest 

 for the money. 



Let r be the amount of 1. for one year ; A, B, C, &c. as in Art. 1, 



then the value required is X I 4- ~^~ 4- -p- + &c. I to the 



end of the tables. 



De Moivre supposes that out of 86 persons born, one dies every year, 

 till they are extinct. On this supposition, the sum of the above series 

 may be found thus. Let n be the number of years which any individual 

 wants of 86 ; then will n be the number of persons living of that age, out 

 of which one dies every year ; then the sum of the above series or the 



- . 



r n 1 



present value of the annuity is -- r - r^ -- = (if P be 



n, (r I) 2 



14 



