436 MALFATTI. 



we see ^ for the probable number of white balls in A after an 



infinite number of operations. Now Malfatti makes Daniel Ber- 

 noulli's statement imply conversely that it will require an infinite 



71/ 



number of trials before the result ^ will probably be reached. 



But Daniel Bernoulli himself does not state or imply this con- 

 verse, so that Malfatti is merely criticising a misapprehension of 

 his own. 



811. Malfatti himself gives a result equivalent to our value 

 of u^ in Art. 809 ; he does not obtain it in the way we use, but 

 by induction founded on examination of successive cases, and not 

 demonstrated generally. 



812. The problem which Malfatti proposes to solve and which 

 he considers analogous to Daniel Bernoulli's is the following. 

 Let r be zero or any given integer not greater than n : required 

 to determine the probability that in x operations the event will 

 never occur of having just n — r white balls in A. This he treats 

 in a most laborious way ; he supposes r = 2, 3, 4, 5 in succession, 

 and obtains the results. He extracts by inspection certain laws 

 from these results which he assumes will hold for all the other 

 values of r between 6 and n inclusive. The cases r — 0, and r = 1, 

 require special treatment. 



Thus the results are not demonstrated, though perhaps little 

 doubt of their exactness would remain in the mind of a student. 

 The patience and acuteness which must have been required to 

 extract the laws will secure high admiration for Malfatti. 



813. We will give one specimen of the results which Malfatti 

 obtains, though we shall adopt an exact method instead of his in- 

 duction from particular cases. 



Required the probability that in x trials the number ?i — 2 of 

 white balls will never occur in A. Let (/> {x, n) represent the whole 

 number of favourable cases in x trials which end with 7i white balls 

 in ^ ; let (ic, n — 1) be the whole number of favourable cases 

 which end with n — 1 white balls in A. There is no other class of 



