MALFATTI. 435 



For the first event a black ball must be taken from A and a white 

 ball from B ; the number of cases is p\ For the second event a 

 black ball must be taken from A and a black one from B, or else 

 a white one from A and a white one from B ; the number of cases 

 is 2p{7i—j)). For the third event a white ball must be taken 

 from A and a black ball from B; the number of cases is 



{n —]pf- 



It is obvious that 



as should be the case. 



809. Now returning to the problem in Art. 807 it will be 

 easy to form the follov/ing equations : 



1 



1 



Integrating these equations and determining the constant by 

 the condition that ^^^ = 1, we obtain 



2 f (- 1)-) 1 j (_ 1)-) 



Daniel Bernoulli's general result for the probable number of 

 white balls in A after x trials if there were 7i originally would be 



Thus supposing x is infinite Daniel Bernoulli finds that the 



71/ 



probable number is ^ . This is not inconsistent with our result ; 



2 1 



for w^e have when x is infinite Vy, = -^y ^^ = t^ > ^iid therefore 



o u 



1 



\ — Vy, — u^—-, so that the case of one white ball and one black 



ball is the most probable. 



810. Malfatti advances an objection against Daniel Bernoulli's 



obtain 

 28—2 



result which seems of no weight. Daniel Bernoulli obtains as 



