THE EVERY-DAY FALLACIES OF GAMBLERS.
THE answer which we publish to-day to our correspondent of last week who described his gambling experience at Saxon- lea-Bains is a curious illustration of the depth to which a com- mon superstition reaches in educated men. It is a popular but very erroneous belief that because it is beforehand much more unlikely that a very uncommon event—say the accidental finding of a sovereign in the street—should happen on two days in succes- sion, than that it should happen once, therefore when it has once happened, it will be much more unlikely for it to happen again at once than it was that it should happen the first time. There is no fallacy which enters deeper into the public mind than this. We have known men who had lost something valuable one day in their walk, go out with quite an easy mind the next, on the ground that it was antecedently so unlikely that they should have such a misfortune two days running, that its occurrence the day before must be considered an insurance against its happening again. People sometimes 'say that a burnt child dreads the fire. And undoubtedly an unlucky Fire Insurance Office dreads a fire. Yet we doubt if even all Insurance- Office Directors are quite free from the false impression that a run of ill-luck against them is less likely after such a run of ill-luck has already taken place, than it would be if the pros- perity for years back had been unclouded. Proverbs like " It's a long lane that has no turning," wjiich are quite true in themselves, are misapplied by the naturally defective logic of the human mind, into arguments why a man might fairly expect the turning to be nearer if he had already walked far without one, than it would be reasonable to suppose it if he had only a very short bit of straight lane behind him. If one person in a family slips on a piece of orange-peel on Monday and breaks his leg, the other members of that family will sometimes go out on Tuesday with a moral certainty of not slipping on pieces of orange-peel and break- ing their legs, because it would be so absurdly unlikely that such an event should happen twice running ; and gamblers habitually act on that extraordinary confusion of ideas,—as modi- fied, however, by another strange superstition which is precisely as unreasonable, namely, that there is a secret tendency now and then to "runs " of luck, so that if for two or three times a particular number comes up in a game of chance, they are half disposed to give Fortune credit for having taken a caprice in its favour, and are disposed to stake upon it for at least once or twice more. Of course, all these .notions are equally groundless. It is no doubt quite true that it is much more improbable that you will find a sovereign in the street two days running, than on one single day ; but the reason why it is more improbable is that the chance of two intrinsically improbable, and yet quite independent, events happen- ing in succession, is compounded of the two chances of each of these events happening, and is therefore much less than either chance separately. But directly one of them has happened, the part of that improbability which is due to the first event of the two is already surmounted, and now the probability of the joint event happening is precisely the same as the probability that the second would happen alone. Our correspondent of this week, who details his experience at Monaco, is evidently not free from the curious confusion of ideas to which we have referred ; though, being ob- viously a thoughtful man, he has struggled to set himself free, and has not yet succeeded. His elaborate account of "La Huitieme Dacompoge " betrays the depth of his superstition, for in point of fact, so far as his theory of play was sound, it had nothing to do with the idea on which it was professedly based ; and so far as it had anything to 4o with that idea, it was unsound. The play he recommends at a game of chance where the chances are even is this :—That you should note down a parti- cular order of eight events, such, for instance, at rouge-et-noir, as " black, black, red, black, black, red, red, red," and assume that as the chance of this compound event happening is only 1 in 256,—which is quite true,—you will have some specially good chance of winning, if you identify your fortunes (in a particular way which we will refer to presently, and which has nothing on earth to do with the theory,) with the particular succession of events, red, red, black, red, red, black, black, black,—in other words, if you substitute the alternative events for those which happened the previous time,—but which, of course, in that order are precisely as unlikely to happen as the first succession. That our correspondent is quite oblivious to this last truth, is evident from the following sentence :—" I once at Monaco won thirteen times run- ning on the octoacheme, without doubling at all, therefore simply following my card like a coffee-mill, until the croupiers them- selves all stared in amazement, their amazement being chiefly due to the fact, not that I won thirteen times, but that I was never strictly with or against the table,' but that my constant success seemed to have nothing to do with so-called runs of luck." Now there the popular superstition comes out quite free from any admixture of true theory. Our correspondent evidently believes that he won " thirteen times in succession," because he was assuming that where red had turned up in the previous runs of eight cards, black was more likely to turn up now, and where black had turned up then, red was more likely to turn up now. That is the nakedest form of the old fallacy to which we have referred. The obvious state of the case was, that whether the croupiers were surprised or not, they had no possible reason for surprise. It was precisely the same kind of run of luck as if red had turned up thirteen times in succession, or black thirteen times in succession, the chance of any one given permutation of red with black being (in a game where chances are even) precisely as unlikely, neither less nor more, than a run of the same number, all red or all black ;—and our correspondent only gave himself needless trouble by selecting one succession of events to favour rather than any other. If he had never noted how the cards had fallen before at all, but had simply staked his money on red or black quite at hazard, on the principle of doubling till he won, and then beginning again with a single stake, he would have had precisely the same chance of winning that he had on the much more elaborate and mind-tasking theory which he was pleased to adopt. The pricking of cards, and all the rest of the ceremonial, was a simple and pure work of supererogation,—one which neither had, nor could have had, the slightest effect on the ultimate chances of winning or losing. What our former correspondent, "An In- structed Gambler," asserted, that there is absolutely no means ever given you by the gambling Banks of staking your money against a compound event, is perfectly true. You are limited to winning to losing upon a simple event of which, in the cane supposed, the chance is assumed to be one-half. There is no mode of selecting red or black at rouge-et-noir, or odd or even at roulette, which gives you any better chance than any other mode. In the long run, it is absolutely indifferent whether you make use of a theory which induces you to stake on odd or on even, on red or on black ; no theory helps in the least, or could help. It is easy enough to make this clear. Suppose that our :correspondent, by consulting his " card," finds that he ought to stake this time on " black," but a new player is come into the room who has not yet pricked on his card the run of former events at all. Will this new player be any the more likely to lose if he stakes on red for want of that former experience which induced our corre- spondent to stake oti black ? It is absurd to say that the game would not be a game of even chances if it were so. a boy who suddenly joins a game of pitch-and-toss has less chance of winning for not having carefully noted, as the former players might have done, how the succession of heads and tails had gone previously. It is obvious that our correspondent's pre- cautions were all naught, and that the theory of the huitieme decomposee is pure rubbish. He would have won just as much or just as little, if he had abandoned his card, and his reference to the eight previous successions of red and black, had put down his money on red or black without even glancing at the colour, only pursuing his rule of always doubling till he won, and then beginning again with a single stake.
For on that part of his theory our correspondent is, of course, quite sound. It is perfectly true that if you can always go on doubling your stake till you win, at any game of chance whatever, —whether the chances be even, or a thousand to one against you, does not matter a button, except as to the capital and the time needed for the game,—you will recover, the first time you win, one stake more than all you have previously lost, and on that you can begin again, as on a new capital. That is a pure arithmetic truth which has really absolutely no more application to one game of chance than to any other, or to any game of chance for that matter, than to any game of skill. The only objection to that course is, first, as our correspondent says, that very few people can afford enough capital to produce what you must produce if you go on thus doubling during a run of twelve or thirteen unfavourable events ; (2) that the banks won't allow you to do this beyond a certain stake ; and (3) that, as far as we know, no Gambling Bank exists in which the chances are allowed to be strictly even, so that you cannot even hope to win on an average once in two trials. Certainly at roulette they are not even, for the 0 is re- garded as neither even nor odd,* the stakes being all pushed aside when the 0 comes up, and confiscated if the next turn of the roulette tables goes against the player's previous choice ; whereas, if it goes in favour of it, they are only restored to the Board, and allowed to take another chance, but the player is not regarded
as having won. This makes the chances of the game really against the player, for whenever a 0 turns up, as of course in a series of only 36 numbers it does not unfrequently, the bank gets a clear advantage. And there is, we believe, a similar disturbing rule at rouge-et-noir, and in the case of all other chances which at first sight seem to be even chances. Hence, even on the very reasonable principle of constantly doubling your stake till you win, you are quite sure to lose in the end, if you go on long enough, either from want of capital, or from the limit put by the bank on the stakes, or from the reserved odds in favour of the table. But all we care to point out is the depth of the fallacy which seems to haunt people on this subject, and which induces an accomplished thinker like our correspondent of to-day to imagine that at a game where the chances are even, he can by some conjuring process of pricking cards somehow outmanceuvre the laws of Chance, and secure more than an even chance for his next risk. It really is nothing but the old fallacy over again,—that as it is very unlikely you should have your pocket picked twice in one day, if you have bad it picked once, you can afford to be more negligent for the rest of the day.
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