372 PROBLEMS ON CAUSES. [CHAP. XX. 



words, past experience does not in this case affect future ex- 

 pectation. 



25. It may be satisfactory to verify this result by ordinary 

 methods. To accomplish this, we shall seek 



First : The probability of drawing r white balls, and p - r 

 black balls, in p trials, out of a bag containing p balls, every ball 

 being replaced after drawing, and all constitutions of the systems 

 being equally probable, a priori. 



Secondly : The value which this probability assumes when 

 ju becomes infinite. 



Thirdly : The probability hence derived, that if m white 

 balls are drawn in succession, the m + I th ball drawn will be 

 white also. 



The probability that r white balls andp - r black ones will be 

 drawn in p trials out of an urn containing /u. balls, each ball 

 being replaced after trial, and all constitutions of the system as 

 above defined being equally probable, is equal to the sum of the 

 probabilities of the same result upon the separate hypotheses of 

 there being no white balls, 1 white ball, lastly ju white balls in 

 the urn. Therefore, it is the sum of the probabilities of this re- 

 sult on the hypothesis of there being n white balls, n varying 

 from to ju. 



Now supposing that there are n white balls, the probability 



of drawing a white ball in a single drawing is - , and the proba- 

 bility of drawing r white balls and p - r black ones in a parti- 

 cular order in p drawings, is 



But there being as many such orders as there are combinations 

 of r things in p things, the total probability of drawing r white 

 balls in p drawings out of the system of /* balls of which n are 

 white, is 



-' 



Again, the number of constitutions of the system of ju balls, which 

 admit of exactly n balls being white, is 



