-476 PHILOSOPHICAL TRANSACTIONS. [aNNO 1788. 



deduced, like the 3 following, from the real probabilities of life. Most of those 

 which are now in use are at best but approximations, and can never be relied on 

 with any tolerable degree of satisfaction. 



Pbob. 1. — Supposing the ages of 2 persons, a and b, to be given; to deter- 

 mine the probabilities of survivorship between them, from any table of obser- 

 vations. 



Solution. — Let a represent the number of persons living in the table at the 

 age of A the younger of the 2 lives. Let a', a", a'", a"", &c. represent the de- 

 crements of life at the end of the 1st, 2d, 3d, 4th, &c. years from the age of 

 A. Let b represent the number of persons living at the age of b the older of 

 the 2 lives, and c, d, e, f, &c. the number of persons living at the end of the 

 1st, 2d, 3d, 4th, &c. years from the age of b. Supposing now it were re- 

 quired to determine the probability of b's surviving a in the first year. It is 

 manifest that this event may take place either by a's dying before the end of the 

 year and b's surviving that period, or by the extinction of both the lives, re- 

 strained however to the contingency of b's having died last. The probability 

 that A dies in the first year, and that b survives it, is expressed by the fraction 



^. The probability that both the lives die in this year is expressed by the 



fraction ^- ^ — ; and as it is very nearly an equal chance that a dies first, this 



fraction should be reduced one-half, and then it will become = ^^-^~—^- Hence 

 the whole probability of b's surviving a in the first year will be = ^ -{- 7 



In like manner, the probability of b's surviving a in the 2d, 3d, 



^o u r A a"Cc + dJ a'" (d -\- e) a"" (e +f) 5 



4th, &c. years may be found = -^-^ .... -~r- — kr" ' ^^- '^ 



spectively; therefore the whole probability of b's surviving a will be = -r 



X (t±^ a: + ^- a" -h ^ a'" + —/■ a"", &c.) Having found, by the pre- 

 ceding series, the probability of b, the elder, surviving a the younger; the 

 other expression, which denotes the probability of a's surviving b, is well known 

 to be the difference between the foregoing series and unity. 



The sum of this series might easily be determined from tables of the ex- 

 pectations of single and joint lives. But no such table as the latter having ever 

 been computed, it will by no means be found a laborious undertaking to com- 

 pute a table of the probabilities of survivorship between 2 persons of all ages 

 immediately from this series, without having recourse to the expectations of 

 life. For if the probability of survivorship between any 2 persons be found, 

 the probability between 2 persons 1 year younger is obtained with little difficulty, 

 and by proceeding in this manner a whole table may be formed in less time than 



