MONTMORT. 87 



lates to games of chance involving cards. The first game is that 

 called Pharaon. 



This game is described by De Moivre, and some investigations 

 given by him relating to it. De Moivre restricts himself to the 

 case of a common pack of cards with four suits ; Montmort sup- 

 poses the number of suits to be any number whatever. On the 

 other hand De Moivre calculates the percentage of gain of the 

 banker, which he justly considers the most important and difficult 

 part of the problem ; see DoctHne of Chances, pages ix, 77, 105, 



Montmort's second edition gives the general results more 

 compactly than the first. 



15i. We shall make some remarks in connection with Mont- 

 mort's investigations on Pharaon, for the sake of the summation of 

 certain series which present themselves. 



155. Suppose that there are p cards in the pack, which the 

 Banker has, and that his adversary's card occurs q times in the 

 pack. Let ii^ denote the Banker's advantage, A the sum of money 

 which his adversary stakes. Montmort shews that 



,. _ g (y - 1) \ A. (p-q){p-q-^) ,, 



;'^-^.0;-l) 2^+ p[p-l) ^'-^ 



supposing that j9 — 2 is greater than q. That is Montmort should 



3 



have this; but he puts {pq — q^) 2 A + {(f — q)-^A, on his page 89, 



1 



by mistake for q^q — l) - A) he gets right on his page 90. Mont- 



mort is not quite full enough in the details of the treatment of 

 this equation. The following results will however be found on 

 examination. 



If q is even we can by successive use of the formula make ?/^ 

 depend on u^ ; and then it follows from the laws of the game that 



Wj is equal io A \i q is equal to 2, and to ^ ^ if ^ is greater 



than 2. Thus we shall have, if q is an even number greater 

 than 2, 



