476 LAPLACE. 



890. Laplace's next memoir on our subject is entitled Me- 

 vioire sur les Prohahilites ; it is contained in the volume for 1778 

 of the Histoire de l Acad.... Paris: the date of publication of the 

 volume is 1781. The memoir occupies pages 227 — 382. 



In the notice of the memoir which is given in the introductory 

 part of the volume the names of Bayes and Price are mentioned. 

 Laplace does not allude to them in the memoir. See Art. 540. 



891. Laplace begins with remarks, similar to those which we 

 have already noticed, respecting the chances connected with the 

 tossing of a coin which is not quite symmetrical; see Arts. 877, 881. 

 He solves the simple problem of Duration of Play in the way we 

 have given in Art. 107. Thus let p denote A's skill, and 1 —p de- 

 note ^'s skill. Suppose A to start with m stakes, and B to start 

 with n — m stakes : then ^'s chance of winning all ^'s stakes is 



P^-ii-pY ' 



1 1 



Laplace puts for p in succession - (1 + a) and « (1 ~ ^)> ^^^ 

 takes half the sum. Thus he obtains for ^'s chance 



|{(l + a)"-+(l-a)"-"j{(l+ar-(l-ar} 



(1 + ay - (1 - a)" ' 



which he transforms into 



1 ln_a=)™ (! + «)""'"'- (!-«)" 



-2m 



2 2^ ^ (l+a)"-(l-a)'' 



oil 

 The expression for ^'s chance becomes — when a vanishes ; 



Laplace proposes to shew that the expression increases as a in- 

 creases, if 2m be less than ??. The factor (1 — a^)"" obviously dimin- 

 ishes as a increases. Laplace says that if 2m is less than n it is 

 clear that the fraction 



(I + ay -{I -ay 



