47^ LAPLACE. 



ou (2) de (3), ou (3) de (4), ... on (n-1) de (71), ou (n) de (1). Be plus, 

 la probabilite d'lin quelconque des joueurs, pour gagner I'autre =^, et 



celle de ne gagner ni perdre =^. Cela pose, il faut determiner la pro- 



babilite que Tun de ces joueurs gagnera la partie au coup x. 



This problem is rather difficult; it is not reproduced in the 

 Theorie...des Proh. The following is the general result: Let v^ 

 denote the chance that any assigned player will win. the match 

 at the ic*^^ trial ; then 



n n {n — 1") 1 n [n — 1) {n — 2) 1 



'^x o ^x-\ "T -j 4> q^ ^X-2 ' -j 9 O 03 ^X-^ I • • • 



1 



syii ^x—n' 



884^. Laplace next takes the Problem of Points in the case 

 of two players, and then the same problem in the case of three 

 players ; see Art. 872. Laplace solves the problem by Finite Differ- 

 ences. At the beginning of the volume which contains the memoir 

 some errata are corrected, and there is also another solution indi- 

 cated of the Pr^^blem of Points for three players; this solution 

 depends on the expansion of a multinomial exj^ression, and is 

 in fact identical with that which had been given by De Moivre. 



Laplace's next problem may be considered an extension of the 

 Problem of Points; it is reproduced in the Theorie...des Proh. 

 page 214, beginning with the words Concevons encore. 



885. The next two problems are on the Duration of Play; in 

 the first case the capitals being equal, and in the second case 

 unequal; see Art. 8G3. The solutions are carried further than in 

 the former memoir, but they are still much inferior to those 

 which were subsequently given in the TIieor{e...des Proh. 



886. The next problem is an extension of the problem of 

 Duration of Play with equal capitals. 



It is supposed that at every game there is the chance ^? for 

 A, the chance q for J3, and the chance r that neither wins; each 

 player has m crowns originally, and the loser in any game gives 

 a crown to the winner : required the probability that the play 

 will be finished in x games. This problem is not reproduced in 

 the Theorie...des Proh. 



