PASCAL AND FERMAT. 9 



posed that the players have equal chances of whining a single 

 point. 



12. We will now give an account of Pascal's investigations 

 on the Problem of Points ; in substance we translate his words. 



The following is my method for determining the share of each 

 player, when, for example, two players play a game of three points 

 and each player has staked 32 pistoles. 



Suppose that the first player has gained two points and the 

 second player one point ; they have now to play for a point on 

 this condition, that if the first player gains he takes all the money 

 which is at stake, namely 6^ pistoles, and if the second player 

 gains each player has two points, so that they are on terms of 

 equality, and if they leave off playing each ought to take 32 

 pistoles. Thus, if the first player gains, 64 pistoles belong to 

 him, and if he loses, 32 pistoles belong to him. If, then, the 

 players do not wish to play this game, but to separate without 

 playing it, the first player w^ould say to the second " I am certain of 

 32 pistoles even if I lose this game, and as for the other 32 pistoles 

 perhaps I shall have them and perhaps you will have them ; the 

 chances are equal. Let us then divide these 32 pistoles equally 

 and give me also the 32 pistoles of which I am certain." Thus 

 the first player wdll have 48 pistoles and the second 16 pistoles. 



Next, suppose that the first player has gained two points and 

 the second player none, and that they are about to play for a 

 point ; the condition then is that if the first player gains this 

 point he secures the game and takes the 64 pistoles, and if the 

 second player gains this point the players will then be in the 

 situation already examined, in which the first player is entitled 

 to 48 pistoles, and the second to 16 pistoles. Thus if they do not 

 wish to play, the first player would say to the second " If I gain 

 the point I gain 64 pistoles ; if I lose it I am entitled to 48 

 pistoles. Give me then the 48 pistoles of which I am certain, 

 and divide the other 16 equally, since our chances of gaining the 

 point are equal." Thus the first player will have 56 pistoles and 

 the second player 8 pistoles. 



Finally, suppose that the first player has gained one point and 



