202 NICOLE. 



This supposes tliat if each wins four games, neither receives 

 nor loses any thing. Now it is obvious that the numerator of the 

 expression is divisible by a + h ; thus we may simplify the ex- 

 pression to 



This is precisely the expression we should have if the players 

 had agreed to play seven games instead of eight. Nicole notices 

 this circumstance, and is content with indicating that it is not 

 unreasonable ; we may shew without difficulty that the result is 

 universally true. Suppose that when A and B agree to play 

 2n — l games, p^ is the chance that A beats B by just one game, 

 ^2 the chance that A beats B by two or more games ; and let 

 ^j, q^ be similar quantities with respect to B, then ^'s advantage 

 is S {p^-\- p^ — q^ — q^. Now consider 2n games : ^'s chance of 



beating B by two or more games, is j?2 + ? ; -S's chance of 



beating A by two or more games is q^ + ^^ . Hence A's ad- 

 vantage is 



S(p.+ 



^2- 



a + b ^2 a+ bj ' 

 Now we know that ^ = ^ = fj, say; therefore 



a 



p^a-qj) ^{a ~b) . 



a + b a + b ^^ i Lx i\ 



Hence the advantage of A for 2/^ games is the same as for 

 2/1 — 1 games. 



853. In the same volume of the Hist, de TAcad....Pa7^is, on 

 pages 331 — 344, there is another memoir by Nicole, entitled 

 Methode jwur determiner le sort de taut de Joueurs que Von 

 voudra, et Vavantage que les iins out sur les autres, lorsqitils 

 joilent h qui gagnera le plus de parties dans un nomhre de parties 

 determine. 



This is the Problem of Points in the case of any number of 

 players, supposing that each player wants the same number of 



