d'alembeht. 287 



divers Savans... in which Laplace had made the supposition that 

 the coin has a gTeater tendency to fall on one side than the other, 

 but it is not known on which side. Suppose that 2 crowns are to 

 be received for head at the first trial, 4 for head at the second, 

 8 for head at the third, . . . Then Laplace shews that if the game is 

 to last for X trials the player ought to give to his antagonist less 

 than X crowns if x be less than 5, and more than x crowns if x be 

 greater than 5, and just x crow^ns if x be equal to 5. On the com- 

 mon hj^pothesis he would always have to give x crowns. These 

 results of Laplace are only obtained by him as approximations ; 

 D'Alembert seems to present them as if they were exact. 



530. Suppose the probability that head should fall at first to 



be ft) and not ^ ; and let the game have to extend over n trial s 



Then if 2 crowns are to be received for head at the first trial, 4 

 for head at the second, and so on ; the sum which the player 

 ouQ:ht to orive is 



2(o ;i + 2 (1 - ft)) + 2^ (1 - ft>)^ + ... + 2"-^ (1 - ft))"-'}, 



which we will call H. 



D'Alembert suggests, if I understand him rightly, that if we 

 know nothing about the value of co we may take as a solution of 



the problem, for the sum which the player ought to give I fldo). 



But this involves all the difficulty of the ordinary solution, for the 

 result is infinite wdien n is. D'Alembert is however very obscure 

 here ; see his pages 45, 46. 



He seems to say that I Cldco will be greater than, equal to, or 



•^ 



less than ??, according as n is greater than, equal to, or less than 5. 

 But this result is false ; and the argument unintelligible or incon- 



elusive. We may easily see by calculation that I D-dw = n when 



n = l\ and that for any value of n from 2 to 6 inclusive 



I Hc^ft) is less than n ; and that when n is 7 or any greater number 







1 



I Q.du> is greater than n. 



' 



