Chance and Luck

In previous pages I have considered, under the head of 'Gamblers' Fallacies,' certain plans by which some fondly imagine that fortune may be forced. I have shown how illusory the schemes really are which at first view appear so promising. There are other plans the fallacy in which cannot be quite so readily seen, though in reality unmistakable, when once the conditions of the problem are duly considered.

Let me in the first place briefly run through the reasoning relating to one of the simpler methods already considered at length.

The simplest method for winning constantly at any such game as rouge-et-noir is as follows: -- The player stakes the sum which he desires to win, say 1£. Either he wins or loses. If he wins he again stakes 1£, having already gained one. If, however, he loses, he stakes 2£. If this time he wins, he gains a balance of 1£, and begins again, staking 1£, having already won 1£. If, however, he loses the stake of 2£, or 3£ in all (for 1£ was lost at the first trial), he stakes 4£. If he wins at this third trial, he is 1£ to the good, and begins again, staking 1£ after having already won 1£. If, however, he loses, he stakes 8£. It will readily be seen that by going on in this way the player always wins 1£ when at last the right color appears. He then, in every case, puts by the 1£ gained and begins again.

It seems then at first as though all the player has to do is to keep on patiently in this way, starting always with some small sum which he desires to win at each trial, doubling the stake after each loss, when he pockets the amount of his first stake and begins again. At each trial the same sum seems certainly to be gained, for he cannot go on losing for ever. So that he may keep on adding pound to pound, ad infinitum, or until the 'bank' tires of the losing game.

The fallacy consists in the assumption that he cannot always lose. It is true that theoretically a time must always come when the right color wins. But the player has to keep on doubling his stake practically, not theoretically; and the right color may not appear till his pockets are cleared. Theoretically, too, it is certain that be the sum at his command ever so large, and the stake the bank allows ever so great, the player will be ruined at last at this game, if -- which is always the case -- the sum at the command of the bank is very much larger. It would be so even if the bank allowed itself no advantage in the game, whereas we know that there is a certain seemingly small, but in reality decisive, advantage in favor of the bank at every trial. Apart from this, however, the longest pocket is bound to win in the long run, at the game of speculation which I have described. For, though it seems a tolerably sure game, it is in reality purely speculative. At every trial there is an enormous probability in favor of the player winning a certain insignificant sum; but, per contra, there is a certain small probability that he will lose, not a small sum, or even a large sum, but all that he possesses -- supposing, that is, that he continues the game with steady courage up to that final doubling which closes his gambling career, and also supposing that the bank allows the doubling to continue far enough; if the bank does not, then the last sum staked within the bank limit is the amount lost by the player, and, though he may not be absolutely ruined, he loses at one fell swoop a sum very much larger than that insignificant amount which is all he can win at each trial.