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the impounded stakes has been decided.
Thus, whereas, with regard to games decided at the first trial, the bank has in the long run no interest one way or the other, the bank has an exceptional interest in refaits. A refait trente-et-un at once gives the bank a certainty of winning the least sum staked in the two compartments, and an equal chance of winning the larger sum instead. Any refait gives the bank the chance that on a new trial a refait trente-et-un may be made; and though this chance (that is, the chance that there will first be a common refait and then a re-fait trente-et-un) is small, it tells in the long run and must be added to the advantage obtained from the chance of a refait trente-et-un at once.
Now it may seem as though the bank would gain very little from so small an advantage. A refait may occur tolerably often in any long series of trials, but a refait trente-et-un only at long intervals. It is only one out of ten different refaits, which to the uninitiated seem all equally likely to occur; so that he supposes the chance of a refait trente-et-un to be only one-tenth of the chance (itself small at each trial) that there will be a refait of some sort. But, to begin with, this supposition is incorrect. Calculation shows that the chance of a refait of some sort occurring is 1,097 in 10,000, or nearly one in nine. The chance of a refait trente-et-un is not one-tenth of this, or about 110 in 10,000, but
219 in 10,000, or twice as great as the uninitiated imagine. Thus in very nearly two games in 91, instead of one game in 91, a refait trente-et-un occurs. It follows from this, combined with the circumstance that on the average the bank wins half its stakes only in the case of one of these refaits (and account being also taken of the slight subordinate chance above mentioned), that the mathematical advantage of the bank is very nearly one-ninetieth of all the sums deposited. The actual percentage is 1 1/10 per deposit, or 1l. 2s. per 100/. And in passing it may be noticed as affording good illustration of the mistakes the uninitiated are apt to make in such matters, that if instead of the refait trente-et-un the bankers took to themselves the refait quarante, then, instead of this percentage per deposit, the percentage would be only 3/20, or 3s. per 100/.
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