Vlll APPENDIX THE FIRST. 



refait* trente et un being '0219* Opposite to D. G. is 

 the sum of the chances of all the refaits except the first, 

 which sum is the chance of a drawn game. 



If then we say that *021 is the chance of a refait 

 trente et un and *087that of a drawn game, there remains 

 892 for the chance that either the bank or the player 

 must win ; which chances being equal, give '446 for 

 the player, and the same for the bank (exclusive of the 

 benefit of the apres). Returning then to the result in 

 page v., we find a = /3='446, 0r='021, and 2a to 2a 

 + is -892 to '913, or 892 to 913. Consequently; 



At the game of rouge et noir, as now played, the 

 chances of ultimate ruin to the bank or the player are 

 the same as they would be at a simple game, which 

 must be either won or lost at each throw, and in which 

 the bank has 913 chances of winning, where the player 

 has 892. 



The bank, as noticed in page 1 1 0., is playing against 

 the whole public, or against a player with unlimited 

 means. Taking 892 to 913, or more correctly 8903 

 to 9122, and applying the rule in page 110., the follow- 

 ing table results, which must be thus used. Opposite 

 to 30 we find *4824, which is the chance of the ultimate 

 ruin of a bank which risks one-thirtieth of its means at 

 every game : 



