CHAP. XX.] PROBLEMS ON CAUSES. 373 



1.2 ..n 



and the number of possible constitutions of the system is 2 

 Hence the probability that exactly n balls are white is 



Multiplying (1) by this expression, and taking the sum of the 

 products from n = to n = ^ we have 





for the expression of the total probability, that out of a system 

 of fj. balls of which all constitutions are equally probable, r white 

 balls will issue in p drawings. Now 



.,.,, ( A .-l)..Q-n+l)/n\Y n\f- T 



- 1.2..W.2" V M/ 



Z> standing for the symbol ^, so that ^ (D) c nd = (w) t n0 . But 

 by a known theorem, 



m 



_ 2). 



In the second member let = #, then 



// A 2 m //* 



^- + =- X* -3- + &C.) (1 -f X>, 

 u37 1 . 2t CtX 



snce 



