X APPENDIX THE FIRST. 



un, and its consequences, is gained on the whole of the 

 opponent stakes. The following is the method of esti- 

 mating the mathematical advantage of the bank : Before 

 a common game, the prospects of the bank lie entirely 

 in the chance of that game being followed by an apres, 

 since, in all other respects, the chances of gain and loss 

 are the same. After a refait trente et un, on whichever 

 colour any player may choose to risk his impounded 

 stake, the bank has the chance a of winning that stake, 

 and none of losing. But besides this, the bank has all 

 the chances of a second refait (or + c) or 1 2a), for 

 having another trial of the same kind : if then x express 

 the fraction of a pound, which such a chance of 11. is 

 worth, we have 



1 2a) r, or 



The mathematical advantage of the bank is therefore, 

 the chance 0, or '0219, of being put in possession of the 

 worth of half the stakes; or 'Oil/, of all the sums 

 it deposits. This is \l. 2s. per cent.* per deposit; 

 which, to those who know the rapidity with which the 

 risks succeed one another, will appear to yield, in the 

 course of the year, an ample return, not merely to the 

 deposits, but to the sums which are reserved for security 

 against fluctuation. It is probably 4J per cent, upon every 

 real risk ; and the return in the course of a year may be 

 easily guessed at. Imagine 100 different games, played 

 on each of 100 different evenings, the sum risked by the 

 collective players on each game being 50/. The total de- 

 posits of the bank would be 500,000/., on which II. 2*. 

 per cent, is 55001. The capital required to make this spe- 

 culation much more safe than any mercantile adventure, 

 would not be larger than its probable return in one year. 



* Some time after this was written I chanced to find the following 

 sentence in the lately published Theory of Probabilities of M. Poisson. 

 " Dans les jeux publics de Paris 1'avantage a chaque coupest peu consider- 

 able : au jeu de trente-et-quarante par exemple, il est un peu au-dessous 

 de "Oil de chaque mise. Voyez sur les chances de ce jeu, le memoire que 

 j'ai inseredans le journal de M. Gergonne, tome xvi. numero 6, Decembre, 

 1825." I have not seen this memoir ; the accordance of the result with my 

 own, shows that I hav described the game correctly, as it was played in 

 Paris j of which, from paucity of information, I was by no means sure. 



