122 MONTMOET. 



Nicolas Bernoulli also indicates another method ; he resolves 

 ^(f + l)finto 



r (r+l)(r + 2) (r + 3) _ r (r + 1) (r+2) 7- (r + 1) 

 1.2.3.4 1.2.3 "^ 1.2 ' 



and thus finds that the required sum is 



??0^4-l) 02+2) (M+3)(n + 4) __ ^ (72 + 1) (w + 2) (?z + 3) 

 1.2.3.4.5 1.2.3.4 



w (^ 4- 1) (w + 2) 



4- 



1.2.3 



208. It seems probable that a letter from Montmort to 

 Nicolas Bernoulli, which has not been preserved, preceded this 

 letter from Nicolas Bernoulli. For Nicolas Bernoulli refers to the 

 problem about a lottery, as if Montmort had drawn his attention 

 to it ; see Art. 180 : and he intimates that Montmort had offered 

 to undertake the printing of James Bernoulli's unpublished Ars 

 Conjectandi. Neither of these points had been mentioned in 

 Montmort's preceding letters as we have them in the book. 



209. The next letter is from Montmort to Nicolas Bernoulli ; 

 it occupies pages 315 — 323. The most interesting matter in this 

 letter is the introduction for the first time of a problem which has 

 since been much discussed. The problem was proposed to Mont- 

 mort, and also solved, by an English gentleman named Waldegrave ; 

 see Montmort's pages 318 and 328. In the problem as originally 

 proposed only three players are considered, but we will enunciate 

 it more generally. Suppose there are n-{-l players ; two of them 

 play a game ; the loser deposits a shilling, and the winner then 

 plays with the third player ; the loser deposits a shilling, and 

 the winner then plays with the fourth player ; and so on. The 

 player who lost the first game does not enter again until after the 

 {n -\-iy^ player has had his turn. The process continues until 

 one player has beaten in continued succession all the other players, 

 and then he receives all the money which has been deposited. 

 It is required to determine the expectation of each of the players, 

 and also the chance that the money will be won when, or before, 

 a certain number of games has been played. The game is sup- 



