ULTIMATE RESULTS OF PLAY. V 



A and B play at a game which presents four cases, 



A, B, D, and T, of which the chances are a, (3, $, and d, 

 so that a-J-/3-l--f-0=l. When A happens., the player 

 A wins ; when B happens, the player B wins ; when D 

 happens, the game is unconditionally drawn ; but when 

 T happens, the game is drawn, and in the next game 

 only the player A puts down a stake, and not the player 



B. If D should follow T, or if T should happen any 

 numher of times running, or D, or successions of T and 

 D, still A's stake remains risked, without any from B : 

 "nor does B stake again, until the happening of A or B 

 recovers A's stake, or assigns it to B : after which, both 

 parties stake again. 



Supposing the means of B to be unlimited, let A m 

 represent A's chance of winning indefinitely, immediately 

 before a game in which both are to stake, A having m 

 stakes in his possession: and let K' m represent A's chance 

 immediately before a game in which B does not stake. 

 Then, by the preceding method 



A m = aA m + i ,_j_ /3 A m __ i -|_ SA m _j- 0A' m 

 A' OT = A m + /3 A m _ i 

 Eliminate h' m and we have 



~ (a + p)*+e& (a+j3 



in which the sum of the two co-efficients is unity. 

 Hence this game is equivalent in its ultimate chances to 

 a simple game in which it is a (a -f /3) to /3 (a + /3) -f 6 ft 

 for A against B. If a=/3, the last odds are those of 

 2a to 2a + 6. 



This game of rouge et noir is described in an unin- 

 telligible manner, and with material omissions, in the 

 later editions of Hoyle, from which work, and from the 

 testimony of persons who have seen it played, I give the 

 best description I can make of it, observing that the 

 most modern method of playing differs in several parti- 

 culars from that given in the book referred to. 



A number of packs of cards is taken (six, it is said in 



