538 LxlPLACE. 



The coefficient of f in the expansion of u in powers of t gives 

 the probabiHty that the play will terminate at the x^^ game. 



The probability that the play will terminate at or before the 

 x^^ game will be the sum of the coefficients of f and of the inferior 

 jDOwers of t in the expansion of u, which will be equal to the co- 



u 



efficient of f in the expansion of- ; that is, it will be the co- 

 efficient of f in the ex^Dansion of 



1 f (2 - t) 



This expression is equal to 



1 r(2-o f f f r \ 



¥ {l-tf \ T (1 - "^ 2'^' (1 - tf 2^''(l-/)-^'^ ""J ' 

 The T^^ term of this development is 



(- 1)*--^ (2 - 1) r 



9r/i, 



r+l > 



that is 



( 1) l^™-! (^1 





ty^^ ^rn ^^ _ ^y^lj • 



The expansion in powers of t of this r* term may now be 

 readily effected ; the coefficient of f will be 



f 1 \ic-\-r — rii \ \x -\-r — rii — 1 

 (- ) I 2^^ \x-rn \r ~ W' \x-rn-\ [T j ' 



/ 2y-i \x -\-r — rn — 1 



that is ^-TTri— , {x — rn + 2r). 



^"^"^ x — rn\r ^ ^ 



The final result is that the probability that the play will termi- 

 nate at or before the x^^ game, is represented by as many terms 

 of the following series as there are units in the integer next 



below - : 

 n 



1 2 3~2^ ^ - o;i + uj — . . . 



