45 G PEEVOST AND LHUILIER. 



This result had been noticed by Condorcet ; see page 189 of 

 the Essai... de l' Analyse... 



845. Out of the r + s + 1 cases considered in the preceding 

 Article, suppose we ask which has the greatest probability ? This 

 question is answered in the memoir approximately thus. A quan- 

 tity when approaching its maximum value varies slowly ; thus we 

 have to find when the result at the end of Article 843 remains 

 nearly unchanged if we put r — 1 for r and s + 1 for s. This 

 leads to 



p + r (7 + 5 + 1 , 

 = z. — , nearly ; 



therefore - = — ~- nearly. 



r s ■\-l ^ 



T 7) 



Thus if r and s are laro^e we have - = - nearly. 



s <i ^ 



846. It will be observed that the expression at the end of 

 Art. 843 is independent of m the number of balls originally con- 

 tained in the urn ; the memoir notices this and draws attention 

 to the fact that this is not the case if each ball is replaced in the 

 urn after it has been drawn. It is stated that another memoir 

 will be given, which will consider this form of the problem when 

 the number of balls is supposed infinite ; but it does not seem that 

 this intention was carried into effect. 



847. It will be instructive to make the comparison between 

 the two problems which we may pi'esume would have formed the 

 substance of the projected memoir. Suppose that j:> white balls 

 have been drawn and q black balls, and not replaced; and suppose 

 the whole number of balls to be infinite : then by Art. 704 the pro- 

 bability that the next r + s drawings will give r white balls and s 

 black balls is 





and on effectinf]^ the intecjration we obtain the same result as in 



