BUFFON. 203 



points. Nicole begins in a laborious way; but he sees tliat the 

 chances of the players are represented by the terms in the ex- 

 pansion of a certain multinomial, and thus he is enabled to give 

 a general rule. Suppose for example that there are three players, 

 whose chances for a single game are a, h, c. Let them play a 

 set of three games. Then the chance that A has of winning 

 the whole stake is a + 3a^ (^ + c) ; and similar expressions give 

 the chances of B and (7; there is also the chance ^ahc that the 

 three players should each win one game, and thus no one prevail 

 over the others. 



Similarly, if they play four games, ^'s chance of winning the 

 whole stake is a^ -\-^a {h + c) + 12a^hc\ there is also the chance 

 Wlf that A and B should share the stake between them to the 

 exclusion of G\ and so on. 



But all that Nicole gives was already well known ; see 

 Montmort's page 353, and De Moivi'e's Miscellanea Analytica, 

 page 210. 



854^. In the year 1733 Bufifon communicated to the Academy 

 of Sciences at Paris the solution of some problems in chances. 

 See Hist, de V Acad.... Pains for 1733, pages 43 — 45, for a brief 

 account of them. The solutions are given in Buffon's Essai 

 dArithm^tique Morale, and we shall notice them in speaking 

 of that work. 



So 5. We now return to the work entitled Of the Laws of 

 Chance, the second part of which we left for examination until 

 after an account had been given of De Moivre's works ; see 

 Arts. 78, 88. 



According to the title page this second part is to be attributed 

 to John Ham. 



Although De Moivre is never named, I think the greater part 

 of Ham's additions are taken from De Moivre. 



Ham considers the game of Pharaon in his pages 53 — 73. This 

 I think is all taken from De Moivre, Ham gives the same in- 

 troductory problem as De Moivi^e ; namely the problem which 

 is XI. in De Moivre's first edition, and x. in his third edition. 



In pages 74 — 94 we have some examples relating to the game 

 of Ace of Hearts, or Fair Chance, and to Lotteries. Here we 



