4oJ< PREVOST AND LHUILIER. 



843. The memoir is devoted to the following problem. An 

 urn contains m balls some of which are white and the rest black, 

 but the number of each is unknown. Suppose that p white balls 

 and q black balls have been drawn and not replaced ; required the 

 probability that out of the next r + s drawings r shall give white 

 balls and s black balls. 



The possible hypotheses as to the original state of the urn are, 

 that there were q black balls, or g + 1 black balls, or q-\-2, ... 

 or m — p. Now form the probability of these various hypotheses 

 according to the usual principles. Let 



P^— (qn — q — n -^V) {m — q — n) to p factors, 



§„= (g' 4- ^ — 1) C*/ + w — 2) to q factors ; 



then the probability of the /i*'^ hypothesis is 



P 



where S denotes the sum of all such products as P„^„. Now if 

 this hypothesis were certainly true the chance of drawing r white 

 balls and s black balls in the next r-\-s drawings would be 



'^n'^n 



where 



Bn= {m — q — p — n + 1) {m — q—p — n) to r factors, 



^n = (^ — 1) (« — 2) to 5 factors, 



iV= number of combinations oi m—p— q things r + 5 at a time. 



Thus the whole required probability is the sum of all the 

 terms of which the type is 



We have first to find 2. The method of induction is adopted 

 in the original memoir ; we may however readily obtain X by the 

 aid of the binomial theorem : see Algebra, Chapter L. Thus we 

 shall find 



[p_\q_ [ m + 1 



[p + q + 1 \m — p — q 



