110 ESSAY ON PROBABILITIES. 



limited means, playing against all who choose to enter., 

 that is, playing against unlimited means. It is, 

 therefore, essential to its existence that some mathe- 

 matical advantage should be allowed, even more than 

 is necessary to reproduce the expenses of its ma- 

 nagement. What I have hitherto said on the subject 

 refers to the relation between the bank and the indi- 

 vidual player against it . but considering the former as 

 the antagonist of all who choose to play, it absolutely 

 requires the protection of a mathematical advantage. 

 But having this advantage, it must, in the long run, 

 ruin its individual opponents ; so that bankruptcy to 

 itself, or degradation and suicide to its customers, are 

 the initial conditions of its existence. But since the 

 banks flourish, it is plain that whatever advantage is 

 necessary to their continuance, is really obtained by 

 them ; and I shall now inquire how much this advantage 

 must be in several cases. 



EXAMILE (Rule V.) It is 30 to 29 for the bank 

 upon each game, and the bank stakes the tenth part of 

 its means at every game. What are the chances of its 

 perpetual continuance ? (b = 30, a = 29, n = 10). 



SO 10 = 590,490,000,000,000 

 29 io _. 420,707,233,300,201 



169,782,766,699,799. 



Answer. About 170 to 421, or such a bank would 

 not be likely to last; that is, in the long run, only 170 

 out of 421 such banks would avoid ruin. 



PROBLEM. What is the mathematical advantage 

 which a bank must have, in order that its permanent 

 continuance may have /c to 1 in its favour ; the sup- 

 position being that the bank stakes the nth part of its 

 means at every game? 



RULE. The odds in favour of the bank, on a single 

 game, must be the rith root of 1 -f k to 1. Thus if, 

 judging by the experience of the Parisian* banks, we say 



* These banks were open to the public and to the municipal police. Of 

 the gaming-houses in London, those who know them must speak. The 



