41i TREMBLE Y. 



Primarias quaestiones hie breviter attingere couabor, methodo diluci- 

 dandae imprimis intentiis. 



^QQ. The first problem is the following. A bag contains an 

 infinite number of white balls and black balls in an unknown 

 ratio ; p white balls and q black have been drawn out in ^ + ^ 

 drawings ; what is the chance that m + n new drawings will give 

 m white and w black balls ? 



The known result is 



\m -f- n ^ ' 



m It 



that is, 



1 x^ {1 -xydx 



■^ 



I m + n \)yi-\- p \n-{-q \p -\- q-\-l 



\'m\n \p \q\'m-\-p-\-7i+q-\-l 



Trembley refers to the memoir which w^e have cited in 

 Art. 551, where this result had been given by Laplace ; see also 

 Art. 704. 



Trembley obtains the result by ordinary Algebra ; the investi- 

 gations are only approximate, the error being however inappreci- 

 able when the number of balls is infinite. 



If each ball is replaced after being drawn we can obtain an 

 exact solution of the problem by ordinary Algebra, as we shall see 

 when we examine a memoir by Prevost and Lhuilier ; and of course 

 if the number of the balls is supposed infinite it will be indifferent 

 whether we replace each ball or not, so that we obtain indirectly 

 an exact elementary demonstration of the important result which 

 Trembley establishes approximately. 



767- We proceed to another problem discussed by Trem- 

 bley. A bag is known to contain a very large number of balls 

 which are white or black, the ratio being unknown. In p-\-q 

 drawings p white balls and q black have been drawn. Required 

 the probability that the ratio of the white to the black lies between 

 zero and an assigned fraction. This question Trembley proceeds 

 to consider at great length ; he supposes p) and q very large and 

 obtains approximate results. 



If the assigned fraction above referred to be denoted by 



