152 DE MOIVRE. 



As De Moivre says A, B, (7, ... are "the numbers of Mr. James 

 Bernoulli in his excellent Theorem for the Summing of Powers." 

 See Art. 112. De Moivre refers for the demonstration of the 

 formula to the Supplement to the Miscellanea Analytica, where 

 the formula first appeared. We shall recur to this in speaking of 

 the Miscellanea Analytica. 



277. Problems xxvii. to xxxil. relate to the game of Quad- 

 rille ; although the game is not described there is no difficulty in 

 understanding the problems which are simple examples of the 

 Theory of Combinations : these problems are not in the first 

 edition. 



278. Problem xxxiil. is To find at Pharaon how much it iiT 

 that the Banker gets per Cent of all the Money that is adventured. 

 De Moivre in his Preface seems to attach great importance to this 

 solution ; but it scarcely satisfies the expectations which are thus 

 raised. The player who stakes against the bank is in fact sup- 

 posed to play merely by chance without regard to what would be 

 his best course at any stage of the game, although the previous 

 investigations of Montmort and De Moivre shewed distinctly that 

 some courses were far less pernicious than others. 



The Banker's adversary in De Moivre's solution is therefore 

 rather a machine than a gambler with liberty of choice. 



279. Problem xxxiv. is as follows : 



Supposing A and £ to play together, that the Chances they have 

 respectively to win are as a to 6, and that B obliges himself to set to A 

 so long as A wins without interruption : what is the advantage that A gets 

 by his hand? 



The result is, supposing each to stake one, 



Itj 1 ^ + ^ + FTi)^ + ^TTif + • • • '" *"-^'"'^"™ } ' 



that is, — = — . 







280. Problems xxxv. and xxxvi. relate to the game dis- 

 cussed by Nicolas Bernoulli and Montmort, which is called Treize 

 or Rencontre; see Art. 162. 



