14j pascal and fermat. 



made in the decision by the superfluous trial or trials. Pascal 

 j)uts this point very clearly. 



In the context of the first passage quoted from Leibnitz in 

 Art. 15, he refers to " les belles pensees de Alea, de Messieurs 

 Fermat, Pascal et Huygens, oil Mr. Roberval ne pouvoit ou ne 

 vouloit rien comprendre." 



The difficulty raised by Roberval was in effect reproduced by 

 D'Alembert, as we shall see hereafter. 



18. Pascal then proceeds to apply Format's method to an 

 example in which there are three players. Suppose that the first 

 player wants one point, and each of the other players two points. 

 The game will then be certainly decided in the course of three 

 trials. Take the letters a, h, c and write down all the combinations 

 which can be formed of three letters. These combinations are the 

 following, 27 in number: 



Let A denote the player who wants one point, and B and C the 

 other two players. By examining the 27 cases, Pascal finds 13 

 Avhich are exclusively favourable to A, namely, those in which a 

 occurs twice or oftener, and those in which a, b, and c each occur 

 once. He finds 3 cases which he considers equally favourable to 

 A and B, namely, those in which a occurs once and b twice ; and 

 similarly he finds 3 cases equally favourable to A and C. On the 

 whole then the number of cases favourable to A may be considered 

 to be 13 + f + f, that is 16. Then Pascal finds 4 cases which 

 are exclusively favourable to B, namely those represented by bbb, 

 ebb, bcb, and bbc ; and thus on the whole the number of cases 



