PASCAL AND FERMAT. 13 



sent to Pascal a solution of the Problem of Points depending on 

 combinations. 



Pascal's second letter is dated August 24th, 1654. He says that 

 Fermat's method is satisfactory when there are only two players, 

 but unsatisfactory when there are more than two. Here Pascal 

 was wrong as we shall see. Pascal then gives an example of 

 Fermat's method, as follows. Suppose there are two players, and 

 that the first wants two points to win and the second three points. 

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

 trials. Take the letters a and h and write down all the combina- 

 tions that can be formed of four letters. These combinations are 

 the following, 16 in number : 



Now let A denote the player who wants two points, and B the 

 player who wants three points. Then in these 16 combinations 

 every combination in which a occurs twice or oftener represents a 

 case favourable to A, and every combination in which h occurs 

 three times or oftener represents a case favourable to B. Thus on 

 counting them it will be found that there are 11 cases favourable to 

 A, and 5 cases favourable to B ; and as these cases are all equally 

 likely, -4's chance of winning the game is to -S's chance as 

 11 is to 5. 



17. Pascal says that he communicated Fermat's method to 

 Roberval, who objected to it on the following ground. In the 

 example just considered it is supposed that four trials will be 

 made ; but this is not necessarily the case ; for it is quite possible 

 that the first player may win in the next two trials, and so the 

 game be finished in two trials. Pascal answers this objection by 

 stating, that although it is quite possible that the game may be 

 finished in two trials or in three trials, yet we are at liberty to 

 conceive that the players agree to have four trials, because, even if 

 the game be decided in fewer than four trials, no difference will be 



