MONTilORT. 93 



"We may observe that if we subtract the last expression from 

 unity we obtain the chance that no card is in its right place. Hence 

 if (f> (n) denote the number of cases in which no card is in its right 

 place, we obtain 



162. The game which Montmort calls Treize has sometimes 

 been called Rencontre. The problem which is here introduced for 

 the first time has been generalised and discussed by the following 

 writers : De Moivre, Doctrine of Chances, pages 109 — 117. Euler, 

 Hist, de T Acad.... Berlin, for 1751. Lambert, Kouveaux Memoires 

 de T Acad. ... Berlin, for 1771. Laplace, TJieorie . . . des Proh. 

 pages 217 — 225. Michaelis, Memoire sur la prohahilite du jeu de 

 rencontre, Berlin, 1846. 



163. Pages 148 — 156 of Montmort relate to the game of Bas- 

 sette. This is one of the most celebrated of the old games : it 

 bears a great resemblance to Pharaon. 



As we have already stated, this game was discussed by James 

 Bernoulli, who summed up his results in the form of six tables ; 

 see Art. 119. The most imi^ortant of these tables is in the fourth, 

 which is in effect also reproduced in De Moivre's investigations. 

 The reader who wishes to obtain a notion of the game may con- 

 sult De Moivre's Doctrine of Chances, pages 69 — 77. 



164. James Bernoulli and De Moivre confine themselves to 

 the case of a common pack of cards, so that a particular card, an 

 ace for example, cannot occur more than four times. Montmort 

 however, considers the subject more generally, and gives formulae 

 for a pack of cards consisting of any number of suits. Montmort 

 gives a general formula on his page 153 which is new in his second 

 edition. The quantity which De Moivre denotes by y and puts 

 equal to ^ is taken to be | by Montmort. 



Montmort gives a numerical table of the advantage of the 

 Banker at Bassette. In the second edition some fractions are 

 left unreduced which were reduced to their lowest terms in the 

 first edition, the object of the change being jDrobably to allow 



