76 JAMES BERNOULLI. 



by the laws of tlie game has an advantage over his antagonist. Thus 

 suppose that ^'s chance of winning in the 1st, 3rd, 5th... games is 

 always p, and his chance of losing q) and in the 2nd, 4th, 6th... 

 games suppose that ^'s chance of winning is q and his chance of 

 losing/?. The chance of B is found by taking that of A from 

 unity ; so that B's chance is p or 5' according as ^'s is q or p. 



Now let A and B play, and suppose that the stake is to be 

 assigned to the player who first wins n games. There is however to 

 be this peculiarity in their contest : If each of them obtains n — 1 

 games it will be necessary for one of them to win two games in 

 succession to decide the contest in his favour; if each of them 

 wins one of the next two games, so that each has scored n games, 

 the same law is to hold, namely, that one must win two games in 

 succession to decide the contest in his favour ; and so on. 



Let us now suppose that n = 2, and estimate the advantage of 

 A. Let X denote this advantage, >S^ the whole sum to be gained. 



Now A may win the first and second games ; his chance for 

 this \^ pq, and then he receives S. He may win the first game, 

 and lose the second ; his chance for this is p^. He may lose the 

 first game and win the second; his chance for this is ^. In the 

 last two cases his position is neither better nor worse than at first ; 

 that is he may be said to receive x. 



Thus X = pq S -{■ {p"^ -{- q^) X \ 



r pq S pq S S 



therefore a?= ., ^ ., 2= ^ =7T • 



1 —p — q zpq A 



Hence of course J5's advantage is also - . Thus the players 



are on an equal footing. 



James Bernoulli in his way obtains this result. He says that 

 whatever may be the value of n, the players are on an equal foot- 

 ing ; he verifies the statement by calculating numerically the 

 chances for n = 2, 8, 4 or 5, taking^ = 2q. See his pages 18, 19. 



Perhaps the following remarks may be sufficient to shew that 

 whatever n may be, the players must be on an equal footing. By 

 the peculiar law of the game which we have explained, it follows 

 that the contest is not decided until one player has gained at least 

 n games, and is at least two games in advance of his adversary. 



