528 LAPLACE. 



whole number of favourable cases to be -^= 1 ; the required 



X \x 

 probability therefore is the latter number divided by the former. 



973, The next problem is the Problem of Points. Laplace 

 treats this very fully under its various modifications ; the dis- 

 cussion occupies his pages 203 — 217. See Arts. 872, 884. 



We will exhibit in substance, Laplace's mode of investigation. 

 Two players A and B want respectively x and y points of winning 

 a set of games ; their chances of winning a single game are jp and 

 q resj^ectively, w^here the sum of "p and q is unity ; the stake is to 

 belong to the player who first makes up his set : determine the 

 probabilities in favour of each player. 



Let </) {x^ y) denote ^'s probability. Then his chance of win- 

 ning the next game is p, and if he wins it his probability becomes 

 </) (a? — 1, y) ; and q is his chance of losing this game, and if he loses 

 it his probability becomes ^{x, y — 1) : thus 



^ {x, y) =p ^ {x -1, y) -\- q (\> {x, y -1) (1). 



Suppose that </> [x, y) is the coefficient of fr^ in the develop- 

 ment according to powers of t and r of a certain function u of 

 these variables. From (1) we shall obtain 



u-t<i>(x, 0)f-t^{0,y) T^+^(0, 0) 



==u(pt + qT)-pttcl>(x,0)f-qTtcl^{0,7j)T' (2), 



where ^ cj) (x, 0) f denotes a summation with respect to x from 

 X = inclusive to x= oo ; and X (/> (0, y) r'" denotes a summation 

 with respect to y from ?/ = inclusive to y= cc . In order to shew 

 that (2) is true we have to observe two facts. 



First, the coefficient of any such term as Tr", where neither m 

 nor n is less than unity, is the same on both sides of (2) by virtue 

 of (1). 



Secondly, on the left-hand side of (2) such terms as Tt", where 

 m or 7i is less than unity, cancel each other ; and so also do such 

 terms on the right-hand side of (2). 



Thus (2) is fully established. From (2) we obtain 



u — , 



1 —j)f — qr 



