PASCAL AND FERMAT. 15 



favourable to B may be considered to be 4 + |, that is 5^. Simi- 

 larly the number of cases favourable to C may be considered to 

 be 5^. Thus it would appear that the chances oi A, B, and C are 

 respectively as 16, 5i, and 51 



Pascal, however, says that by his own method he had found 

 that the chances are as 17, 5, and 5. He infers that the differ- 

 ence arises from the circumstance that in Fermat's method it is 

 assumed that three trials will necessarily be made, which is not 

 assumed in his own method. Pascal was wrong in supposing that 

 the true result could be affected by assuming that three trials 

 w^ould necessarily be made ; and indeed, as we have seen, in the 

 case of two players, Pascal himself had correctly maintained 

 against Roberval that a similar assumption was legitimate. 



19. A letter from Pascal to Format is dated August 29th, 1654. 

 Format refers to the Problem of Points for the case of three 

 players; he says that the proportions 17, 5, and 5 are correct for 

 the example which we have just considered. This letter, how- 

 ever, does not seem to be the reply to Pascal's of August 24th, but 

 to an earlier letter which has not been preserved. 



On the 25th of September Format writes a letter to Pascal, 

 in which Pascal's error is pointed out. Pascal had supposed 

 that such a combination as ace represented a case equally favour- 

 able to A and C\ but, as Format says, this case is exclusively 

 favourable to A, because here A gains one point before C gains 

 one ; and as A only wanted one point the game is thus decided 

 in his favour. When the necessary correction is made, the result 

 is, that the chances of A, B, and C are as 17, 5, and 5, as Pascal 

 had found by his own method. 



Fermat then gives another solution, for the sake of Roberval, 

 in which he does not assume that three trials will necessarilv be 

 made; and he arrives at the same result as before. 



In the remainder of his letter Fermat enunciates some of his 

 memorable propositions relating to the Theory of Numbers. 



Pascal replied on October 27th, 1654, to Fermat's letter, and 

 said that he was entirelv satisfied. 



