LAPLACE. 537 



game should win this, game and the n — 2 following games. Thus 



F' 1 



and therefore ^ z^_^ is the probability that the play will terminate 



at the ic*^ sfame relative to this case. 



By proceeding thus, and collecting all the partial probabilities 

 we obtain 



1 1 )_ 1 n\ 



Suppose that z^ is the coefficient of f in the expansion accord- 

 ing to powers of ^ of a certain function u of this variable. Then 

 from (1) we have, as in Art. 937, 



F(t) 



u = 



111 1 



-*• 2 " 92 '' 2^ * " 2'*"'^ 



where -F (^) is a function of t which is at present undetermined. 



Now if (1) were true for x = n SiS well as for higher values of 

 n, the function F(t) would be of the degree n — 1. But (1) does 

 not hold when x — n, for in forming (1) the player who wins the 

 money was supposed to start against an opponent who had won 

 one game at least ; so that in (1) we cannot suppose x to be less 

 than n-\-l. Hence the function F [t) will be of the degree n, 

 and we may put 



a^ + ait + ctf + . • . + a.f' 



u = 



111 1 



Now the play cannot terminate before the ?i*'' game, and the pro- 

 bability of its terminating at the n^^ game is ^^^ ; therefore a^ 



vanishes for values of x less than n, and a,, = — — • . Thus 



