536 LAPLACE. 



is that which determines the probability that the r^^ player will- 

 win the money precisely at the x^^ game ; Nicolas Bernoulli had 

 confined himself to the probability which each player has of 

 winning the money on the whole. 



982. We will give, after Laplace, the investigation of the 

 probability that the play will terminate precisely at the x^^ 

 game. 



Let z^ denote this probability. In order that the play may 

 terminate at the x^^ game, the player who enters into play at the 

 {x — n-\-Vf^ game must win this game and the n — 1 following 

 games. 



Suppose that the winner of the money starts with a player 

 who has won only one game ; let P denote the probability of this 



P . 



event ; then — will be the corresponding probability that the 



play will terminate at the x^^ game. But the probability that the 

 play will terminate at the {x — iy^ game, that is z^_^, is equal 



P . . 



to ^Tpi . For it is necessary to this end that a player who has 



already won one game just before the {x — n + Vf^ game should 

 win this game and the n — ^ following games j and the probabilities 



of these component events being respectively P and ^^^^i , the 



. P 



probability of the compound event is ^^^ . Thus 



P 1 



— V • 



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



Li 



at the fl?*^ game, relative to this case. 



Next suppose that the winner of the money starts with a player 

 who has won two games ; let P' denote the probability of this 



P' 



event ; then -^ will be the corresponding probability that the play 



P' 



will terminate at the x^"^ game. And -^^ = z^_^ : for in order that 



the play should terminate at the (ic— 2)"* game it is necessary that 

 a player who has already won two games just before the (a; — 72+ 1)"' 



