MONTMORT. 97 



les fautes de raisonnement qui sont dans cette Lettre ; il nous suffira 

 d'avertir que la cause de son erreur est de n'avoir point d'egard aux 

 divers arrangemens des lettres. 



Montmort's words seem to imply that Pascal's letter contains 

 a large amount of error ; we have, however, only the single fun- 

 damental inaccuracy which Fermat corrected, as we have shewTi in 

 Art. 19, and the inference that it was not allowaVjle to suppose 

 that a certain number of trials will necessarily be made; see Art. 18. 



172. Montmort gives for the first time two formulae either of 

 which is a complete solution of the Problem of Points when there 

 are two players, taking into account difference of skill. We will 

 exhibit these formulae in modern notation. Suppose that A wants 

 711 points and B wants n points ; so that the game will be neces- 

 sarily decided in m-\-n—l trials ; \etm + n—l = r. Let p denote 

 A's skill, that is his chance of winning in a single trial, and let 

 q denote J5's skill ; so that p + q = l. 



Then ^'s chance of winning the game is 



pr^ r-i _^ r(r-l) ,_^ + ,— ,^^— fi^V"; 



^"^^ 1.2^ [m I ?? — 1 



and Bs chance of winning the game is 



q'+rr'p+^^-^Y^ 2-p=+ + ^^zi^ ?>"'- ■ 



This is the first formula. According to the second formula J's 

 chance of winning the game is 



m f 1 . m (m + 1) « , , 1/ "" i_ ^K-i 1 . 



and B's chance of winning the game is 



„ f- , 7^ (n + 1) „ , l^~^ ,,--' I 



^ r-^^'^"^ 172 ^+ ^\m-i.n-ij r 



Montmort demonstrates the truth of these formulae, but we 

 need not crive the demonstrations here as they will be found in 

 elementary works; see Algebra, Chapter Llli. 



173. In Montmort's first edition he had confined himself 

 to the case of equal skill and had given only the first formula, 



