OH some Questions in the Theory of Chances. 149 



It will be seen from the formula 



m„ + m„^i m„^^ + q 



m, + m2 + THp +p' 



p. 6. 1. 10. that if life were divided into an infinite number of ages 

 or intervals (in which ease p is infinite), the hypothesis of pos- 

 sibility remaining- the same, the probability of an individual 

 dying in any given interval would be the given interval divided 

 by the whole duration of life, which coincides with that whirli 

 is given by De Moivre's hypothesis. Thus if life were supposed 

 to extend to a hundred years, the probability of an individual 

 dying in any given year would be ^ , and any finite number 

 of observations or recorded deaths would not influence the value 

 of this probability. As diseases, and other causes producing 

 death, are not equally distributed throughout life, the last hy- 

 pothesis cannot be adopted. 



In order to investigate accurately the probability of deatli 

 at any age. it woixld be necessary to know the law of possibility. 

 Let (ppXp be the probability of the possibility of .r^, then the 

 probability in the former question of having n, balls of the first 

 colour, Wj of the second, &c. in ??i + h.j +np trials, is 



fx,'"' {(p,x,) , x"''{<p„X2) (1 — X, - x.2...Xj,_iy"pdx, dx.2 — dx 



_i 



<^ is a sign of function, and this function may be either continuous 

 or discontinuous. 



This expression must be integrated between the same limits 

 as before. 



^ The coefficients of the different powers of x,, in (ppX,,, or the 

 constants in (PpXj,, will generally lie functions of the index p. 

 If the probaliility of life were known at a great many places, 



