MONTMORT. 09 



expresses the chance that A shall win m points before either a 

 single drawn contest occurs, or B wins n points. 



This is easily seen by examining the reasoning by which the 

 formula is established in the case when p -{- q is equal to unity. 



But the formula 



expresses the chance that A shall win m points out of r, on the 

 condition that r trials are to be made, and that A is not to be con- 

 sidered to have won if a drawn contest should occur even after he 

 has won his m points. 



This follows from the fact that if we expand (2^ + q + 1 —p — qY 

 in powers of j^, q, 1 — ^ — 5', a term such as Cj^^q^il —2^ — qy ex- 

 presses the chance that A wins p points, B wins a points, and r 

 contests are drawn. 



Or we may treat this second case by using the transformation 

 in Art. 174. Then we see that {p + qy"^ expresses the chance 

 that there shall be no dra^\Ti contest after the m points which A is 

 supposed to have won ; {p-{- ^)'""'"^ expresses the chance that there 

 shall be no drawn contest after the m points which A is supposed 

 to have won, and the single point which B is supposed to have 

 won ; and so on. 



176. Montmort thinks it might be easily imagined that the 

 chances of A and B, if they respectivel}' want km and Jen points, 

 would be the same as if they respectively wanted m and 71 points ; 

 but this he says is not the case ; see his page 24? 7. He seems to 

 assert that as k increases the chance of the player of greater skill 

 necessarily increases with it. He does not however demonstrate this. 



We know by Bernoulli's theorem that if the number of trials 

 be made large enough, there is a very high probability that the 

 number of points won by each player respectively will be nearly in 

 the ratio of his skill ; so that if the ratio ofm to n be less than that 

 of the skill of A to the skill of B, we can, by increasing k, obtain as 

 great a probability as we please that A will win km points before 

 B wins hi points. 



Montmort probably implies, though he does not state, the con- 

 dition which we have put in Italics. 



7—2 



