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es. He puts down 21. under M and L, and, following the simple rule, stakes 41. Say he wins. He then puts Sown 4l. under w, and scores out 21. and 21., the only two remaining numbers under M. A, therefore, now closes his little account, finding himself the winner of
81., 11/., 91., and 41., or 32l. in all, and the loser of 2l., 51., 51., 8/., and 2/., or 22/. in all, the balance in his favor being 101., the sum he set forth to win. It seems obvious that the repetition of such a process as this, any convenient number of times at each sitting, must result in putting into A's pocket a considerable number of the sums of money dealt with at each trial. In fact, it seems at a first view that here is a means of obtaining untold wealth, or at least of ruining any number of gambling, banks.
Again, at a first view, this method seems in all respects an immense improvement on the simpler one. For whereas in the latter only a small sum can be gained at each trial, while the sum staked increases after each failure in geometrical progression, in this second method (though it is equally a gambling superstition) a large sum may be gained at each trial, and the stakes only increase in arithmetical progression in each series of failures. The comparison between the two plans comes out best when we take the sum to be won undivided, when also the system is simpler; and, further, the fallacy which underlies this, like every system for gaining money with certainty, is more readily detected, when we consider it thus.
Take, then, the sum of 101., and suppose 51. the first loss, after which take two losses, one gain, one loss, and two gains. The table will be drawn up then as shown--with the balance of 10/., according to the
fatals success of this system.
On the other hand, take the other and simpler method, where we double the original stake aider each
failure. Then supposing the losses and gains to follow in the same succession as in the case just considered, note that the first gain doses the cycle. The table has the following simple form (counting three losses to begin with):
We see then at once the advantage in the simpler plan which counterbalances the chief disad-
vantage mentioned above. This disadvantage, the rapid increase of the sum staked, is undoubtedly serious; but, on the other hand, there is the important advantage that at the first success the sum originally staked is won; whereas, according to the other plan, every failure puts a step between the player and final success. It can readily be shown that this disadvantage in the less simple plan just balances the disadvantage in the simpler plan.
But now let us more particularly consider the probabilities for and against the player involved in the plan we are dealing with.
Note in the first place that the player works down the column under M from the top and bottom, taking off two figures at each success, and each figure adding one figure at the bottom after each failure. To get
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