530 LAPLACE. 



«. 



p T 



coefficient of f the expression _ J^ .^ _ y . Then expand 



this expression according to powers of r, and we finally obtain for 

 the coefficient of fj^ 



This is therefore the probability in favour of A ; and that in 



favour of JB may be obtained by interchanging p with q and x 



with ?/. 



The result is identical with the second of the two formuloe 



which we have given in Art. 172. 



97*i. The investigation just given is in substance Laplace's ; 



he takes the particular case in which p = -^ and q = ^', but this 



makes no difference in principle. But there is one important 

 difference. At the stage where we have 



F{t) +/(t) 



u = 



1 —pt — qr 



Laplace puts 



fir) 



u = 



1 —jyi — qr 



This is an error, it arises from a false formula given by Laplace 

 on his page 82; see Art. 955. Laplace's error amounts to neg- 

 lectinsc the considerations involved in the second of the facts on 

 which equation (2) of the preceding Article depends : this kind 

 of neglect has been not uncommon with those who have used or 

 expounded the method of Generating Functions. 



975. We will continue the discussion of the Problem of Points, 

 and suppose that there are more than two players. Let the first 

 player want x^ i^oints, the second x^ points, the third x^ points, 

 and so on. Let their respective chances of winning a single game 

 ^^ 1\' 2\> 1\^ • ' • Let cj) (x^, x^, x^, . . .) denote the probability in 

 favour of the first player. Then, as in Art. 973, we obtain the 

 equation 



