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It may appear paradoxical to say, that there is chance that results right themselves— nay, that there is an absolute certainty that in the long run they will occur as often (in proportion) as their respective chances warrant, and at the same time to assert that it is utterly useless for any gambler to trust to this circumstance. Yet not only is each statement true, but it is of first-rate importance in the study of our subject that the truth of each should be clearly recognized.
That the first statement is true, will perhaps not be questioned. The reasoning on which it is based would be too abstruse for these pages; but it has been experimentally verified over and over again. Thus, if a coin bo tossed many thousands of times, and the numbers of resulting "heads" and " tails " be noted, it-is found, not necessarily that these numbers differ from each other by a very small quantity, bat that their difference is •small compared with either. In mathematical phrase, the two numbers are nearly in a ratio of equality. Again, if a die be tossed, say, six million times, then, although there will not probably have been exactly a million throws of each face, yet the number of throws of each face will differ from a million by a quantity very small indeed compared with the total number of throws. So certain is this law, that it has been made the means of determining the real chances for an event, or of ascertaining fscts which had been before unknown. Thus, De Morgan relates the following «ory in illustration of this law. He received it "from a distinguished naval officer, who was once employed to bring home a cargo of dollars." "At the end of the voyage," he Bays, " it was discovered that one of the boxes which contained them had been forced; and on making farther search a large bag of dollars was discovered in the possession of some one on board. The coins in the different boxes were a mixture of all manner of dates and wvereigns; and it occurred to the commander, that if the contents of the boxes were sorted, a comparison of the proportions of the different sorts in the bag, with those in the box which had been opened, would afford presumptive evidence one way or the other. This comparison was accordingly made, and the agreement between the distribution of the several coins in the bag and those in the box, was such as to leave no doubt as to the former baring formed a part of the latter." If the bag of stolen dollars had been a small one, the inference would have been insafe, but
the great number of the dollars corresponded to a great number of chance trials; and as in such a large scries of triuls the several results would be sure to occur in numbers corresponding to their individual chances, it followed that the numbers of coins of the different kinds in the stolen lot would be proportional, or very nearly so, to the numbers of those respective coins in the forced box. Thus in this case the thief increased the strength of the evidence against him by every dollar he added to his ill-gotten store.
We may mention, in passing, an even more curious application of this law, to no less a question than that much talked of, but little understood problem, the squaring of the circle. It can be shown by. mathematical reasoning, that, if a straight rod be so tossed at random into the air as to fall on a grating of equidistant parallel bars, the chance of the rod falling through depends on the length and thickness of the rod, the distance between the parallel bars, and the proportion in which the cirj cumference of a circle exceeds the diameter. So that -when the rod and grating have been carefully measured, it is only necessary to know the proportion just mentioned in order to calculate the chance of the rod falling through. But also, if we ! can learn in some other way the chance of I the rod falling through, we can infer the | proportion referred to. Now the law we : are considering teaches us that if we only toss the rod often enough, the chance of its falling through will be indicated by the number of times it actually does fall 1 through, compared with the total number of trials. Hence we can estimate the proportion in which the circumference of a circle exceeds the diameter, by merely ! tossing a rod over a grating several thousand times, and counting how often it falls through. The experiment has been tried, and Professor De Morgan tells us that a very excellent evaluation of the celebrated proportion (the determination of which ia equivalent in reality to squaring the circle) was the result.
And let it be noticed in passing that this inexorable law — for in its effect it is the most inflexible of all the laws of probability — shows how fatal it must be to contend long at any game of pure chance, where the odds are in favour of our opponent. For instance, let us assume for a moment that the assertion of the foreign gaming bankers is true, and that the chances are but from 11-4 to 21-2 per cent in their favour. Yet in the long run, this percentage must manifest its effects. Where a few hundreds have been wagered the bank may win 1 1-4 or 2 1-2 on each, or may lose considerably; but where thousands of hundreds are wagered, the bank will certainly win about their percentage, and the players will therefore lose to a corresponding extent. This is inevitable, so only that the play continue long enough. Now it is sometimes forgotten that to ensure such gain to the bank, it is by no means necessary that the players should come prepared to stake so many hundreds of pounds. Those who sit down to play may not have a tithe of the sum necessary — if only wagered once — to ensure the success of the bank. But every florin the players bring with them may be, and commonly is, wagered over and over again. There is repeated gain and loss, and loss and gain; insomuch that the player who finally loses a hundred pounds, may have wagered in the course of the sitting a thousand or even many thousand pounds. Those fortunate beings who "break the bank " from time to time, may even have accomplished the feat of wagering millions during the process which ends in the final loss of the few thousands they may have begun with.
Why is it, then, it will be asked, that this inexorable law is yet not to be trusted? For this reason, simply, that the mode of its operation is altogether uncertain. If in a thousand trials there has been a remarkable preponderance of any particular class of events, it is not a whit more probable that the preponderance will be compensated by a corresponding deficiency in the next thousand trials than that it will be repeated in that set also. The most probable result of the second thousand trials is precisely that result which was most probable for the first thousand — that is that there will be no marked preponderance either way. But there may be such a preponderance; and it may lie either way. It is the same with the next thousand, and the next, and for every such set. They are in no way affected by preceding events. In the nature of things, how they can be Y But "the whirligig of time brings in its revenges" in its own way. The balance is restored just as chance directs. It may be in the next thousand trials, it may be not before many thousands of trials. We are utterly unable to guess when or how it will be brought about.
But it may be urged that this is mere assertion; and many will be very ready to believe that it is opposed to experience, or even contrary to common sense. Yet ex
perience has over and over again confirmed the matter, and common sense, though it may not avail to unravel the seeming paradox, yet cannot insist on the absurdity that coming events of pure chance are affected by completed events of the same kind. If a person has tossed "heads" nine times running (we assume fair and lofty tosses with a well-balanced coin), common sense teaches him, as he is about to make the tenth trial, that the chances on that trial are precisely the fame as the chances on the first. It would indeed have been rash for him to predict that he would reach that trial without once failing to toss "head;" but as the thing has happened, the odds originally against it count for nothing. They arc disposed of by known facts. Wo have said, however, that experience confirms our theory. It chances that a series of experiments have been made on coin-tossing. Buffon was the experimenter, and he tossed thousands of times, noting always how many times he tossed "head" running before "tail," appeared. In the course of these trials he many times tossed "head" nine times running. Now, if the tossing "head " nine times running rendered the chance of tossing a tenth head much less than usual, it would necessarily follow that in considerably more than one half of these instances Buffon would have failed to toss a tenth head. But ho did not. We forget the exact numbers, but this we know, that in about half of the cases in which he tossed nine " heads" running, the next trial also gave him " head;" and about half of these tossings of ten successive "heads" were followed by the tossing of an eleventh "head." In the nature of things this was to be expected.
And now let us consider the cognate questions suggested by our sharper's ideas respecting the person who plays. This person is to consider carefully whether he is " in vein," and not otherwise to play. He is to be cool and businesslike, for fortune is invariably adverse to an angry player. Steinmetz, who appears to place some degree of reliance on the suggestion that a player should be "in vein," cites in illustration and confirmation of the rule the following instance from his own experience : — "I remember," he says, "a curious incident in my childhood which seems very much to the point of this axiom. A magnificent gold watch and chain were given towards the building of a church, and my mother took three chances which were at a very high figure, the watch and chain being valued at more than' One of these chances was entered in my name, one in my brother's, and the third in my mother's. I had to throw for her as well as myself. My brother threw an insigoiBcant figure; for myself I did the same; but, oddly enough, I refused to throw for my mother on finding that I had lost my chance, saying that I should wait a little longer — rather a curious piece of prudence" (read, rather, superstition) "for a child of thirteen. The raffle was with three dice; the majority of the chances had been thrown, and' thirty-four' was the highest." (It is to be presumed that three dice were thrown twice, yet - tbirty-four" is a remarkable throw with til dice, and "thirty-six" altogether exceptional.) "I went on throwing the dice for amusement, and was surprised to find that every throw was better than the one I had in the raffle. I thereupon said, 'Sow I'll throw for mamma.' I threw thirty-six, which won the watch I My mother had been a large subscriber to the building of the church, and the priest said that my winning the watch for her was quite jiroi-idenlial. According to M. Houain'a authority, however, it seems that I only got into • vein,' — but how I came to pause and defer throwing the last chance has always puzzled me respecting this incident of my childhood, which made too great an impression ever to be effaced."
It is probable that most of our readers can recall some circumstance in their lives, tome surprising coincidence, which has caused a similar impression, and which they have found it almost impossible to regard as strictly fortuitous.
In chance games especially, curious coincidences of the sort occur, and lead to tie superstitious notion that they are not mere coincidences, but in some definite way associated •with the fate or fortune of the player, or else with some event which has previously taken place, — as a change of peat?, a new deal, or the like. There is scarcely a gambler who is not prepared to assert 'his faith in certain observances whereby, as he believes, a change of luck may be brought about. In an old work on card-games the player is gravely advised, if the luck has been against him, to turn three times round with his chair, "for then the luck will infallibly change in your favour."
Equally superstitions is the notion that anger brings bad luck, or. as M. Houdin's aothority puts it, that " the demon of bad tack invariably pursues a passionate player." At a game of pure chance good temper makes the player careless under ill-fortune,
but it cannot secure him against it. In like manner, passion may excite the attention of others to the player's losses, and in any case causes himself to suffer more keenly under them, but it is only in this sense that passion is unlucky for him. He is as likely to make a lucky hit when in a rage as in the calmest mood.
It is easy to see ho\v superstitious such as these take their origin. We can understand that since one who has been very unlucky in games of pure chance, is not antecedently Tikely to continue equally unlucky, a superstitious observance is not unlikely to be followed by a seeming change of luck. When this happens the coincidence is noted and remembered; but failures are readily forgotten. Again, if the fortunes of a passionate player be recorded by dispassionate bystanders, he will not appear to be pursued by worse luck than his neighbours; but he will be disposed to regard himself as the victim of unusual ill-fortune. He may perhaps register a vow to keep his temper in future; and then his luck may seem to him to improve, even though a careful record of his gains and losses would show no change whatever in his fortunes.
But it may not seem quite so easy to explain those undoubted runs of luck, by which players" in the vein," (as supposed) have broken gaming-banks, and have enabled those who have followed their fortunes to achieve temporary success. The history of the notorious Garcia, and of others who like him have been for awhile the favourites of fortune, will occur at once to many of our readers, and will appear to afford convincing proof of the theory that the luck of sucli gamesters has had a real influence on the fortunes of the game. The following narrative gives an accurate and graphic picture of the way in which these "bank-breakers" are followed and believed in, while their success seems to last.
The scene is laid in one of the most celebrated German Kursaals.
"What a sudden influx of people into the room I Now, indeed, we shall see a celebrity. The tall light-haired young man coming towards us, and attended by such a retinue, is a young Saxon nobleman who made his appearance here a short time ago, and commenced his gambling career by staking very small sums; but, by the most extraordinary luck, lie was able to increase his capital to such an extent that he now rarely stakes under the maximum, and almost always wins. They say that when the croupiers see him place
his money on the table, they immediately prepare to nay him, without waiting to see which colour lias actually won, and that they offered him a haudsome sum down to desist from playing while he remains here Crowds of people'stand outside the Kursaa' doors every morning, awaiting his arrival and when he cornea following him into the room, and staking as he stakes. When he ceases playing they accompany him to the door, and shower on him congratulation! and thanks for the good fortune he has brought them. See how all the people make way for him at the table, and how deferential are the subdued greetings of his acquaintances! He does not bring much money with him, his luck is too great to require it. He takes some notes out of a case, and places maximums on block and couleur, A crowd of eager hands are immediately outstretched from all parts of the table, heaping up silver and gold and notes on the spaces on which he has staked his money, till there scarcely seems room for another coin, while the other spaces on the table only contain a few florins staked by sceptics who refuse to believe in the count's luck." He wins and the narrative proceeds to describe his continued successes, until ho rises from the table a winner of about one hundred thousand francs at that sitting.
The success of Garcia was so remarkable at times as to effect the value of the shares in the Prieilcrfirte Bank ten or twenty percent. Nor would it be difficult to cite many instances which seem to supply incontrovertible evidence that there is something more than common chance in the temporary successes of these (socalled) fortunate men.
Indeed, to assert merely that in the nature of things there can be no such thing as luck that can be depended on even for a short time, would probably be quite useless. There is only one way of meeting the infatuation of those who trust in the fates of lucky gamesters. We can show that granted a sufficient number of trials, and it will be remembered that the number of those who have risked their fortunes at roulette and rouge et noir is incalculably great — there must inevitably be a certain number who appear exceptionally lucky— or, rather, that the odds are overwhelmingly against tne continuance of play on the scale which prevails at the foreign gambling tables, without the occurrence of several instances of persistent runs of luck.
To remove from the question the perplexities resulting from the nature of the ' tossers, we shall see that it is not merely
above named games, let us suppose that the tossing of a coin is to determine the success-or failure of the player, and that he will win if he throws '• head." Now if a player tossed " head" twenty times running on any occasion it would be regarded as a most remarkable run of luck, and it would not be easy to persiiiide those who witnessed the occurrence that the thrower was not in some special and definite manner the favourite of fortune. We may take such exceptional success as corresponding to tlio good fortune of a "bankbreaker." Yet it is easily shown that with a number of trials which must fall enormously short of the number of cases in which fortune is risked at foreign Kursaals, the throwing of twenty successive heads would be practically ensured. Suppose every adult person in Britain — say 10,000,000 persons in all — were to toss a coin, each tossing until "tail" was thrown; then it is practically certain that several among them would toss twenty times before "tail" was thrown. Thus, it is certain that about five millions would toss
head" once; of these about one half, or some two millions and a half would toss "head" on the second trial; about a million and a quarter would toss
head" on the third trial; about six hundred thousand on the forth; some three hundred thousand on the fifth; and by proceeding in this way — roughly halving the numbers successively obtained — we find that some eight or nine of the ten million persons would be almost certain to toss "head " twenty times running. It must be remembered that so long as the numbers continue large the probability that about half will toss "head" at the next trial amounts almost to certainty. For
xample, about 140 toss •• heads " sixteen times running: now it is utterly unlikely that of these 140, fewer than 60 will tos»
head" yet a seventeenth time. But if the above process failed on trial to give even one person who tossed heads twenty times running — an utterly improbable event — ret the trial could be made four or five limes, with practical certainty that not one or two, but thirty or forty, persons would achieve the seemingly incredible 'eat of tossing " head " twenty times running. Nor would all these thirty or forty icrsons fail to throw even three or four more "heads."
Now if we consider the immense num>er of trials made at gambling tables, and f we further consider the gamblers as in a sense typified by our ten millions of coin
probable but absolutely certain that from time to time there must be marvellous runs of luck at roulette, rouge et noir, hazard, faro, and other games of chance. Suppose that at the public gaming-tables on the continent there eit down each night but one thousand persons in all, that each person makes but ten ventures each nigbt, and that there are but one hundred gambling nights in the year — each supposition falling far below the truth — there are then one million ventures each year. It cannot be regarded as wonderful, then, that among the fifty millions of ventures made (on this supposition) during the last half century, there should be noted some runs of luck which on any single trial would seem incredible. On the contrary, this is so far from being wonderful that it would be far more wonderful if no such runs of luck had occurred. It is probable that if the actual number of ventures, and the circumstances of each, could be ascertained, and if any mathematician could deal with the tremendous array of figures in such sort as to deduce the exact mathematical chance of the occurrence of bank-breaking runs of luck, it would be found that the antecedent odds were many millions to one in favour of the occurrence of a certain number of such events. In the simpler case of our coin-tossers the chance of twenty successive "heads" being tossed can be quite readily calculated. We have made the calculation, and we find that if the ten million persons had each two trials the odds would be more than 10,000 to 1 in favour of the occurrence of twenty successive "heads " once at least; and only a million and a half neud have a single trial each, in order to give an even chance of luch an occurrence.
But we may learn a further lesson from our illustrative tossers. We have seen that granted only a sufficient number of trials, runs of luck are practically certain to occur; but we may also infer that no run of luck can be trusted to continue. The very principle which has led us to the conclusion that several of our tossers would throw twenty " heads "successively, leads also to the conclusion that one who has tossed beads twelve or thirteen times, or any other considerable number of times in succession, is not more (or less) likely to toss " head " on the next trial than at the beginning. About half, we said, in discussing the fortunes of the tossers, would toss •• head " at the next trial: in other words, about half would fail to toss " head." The chances for and against these lucky tossers are equal at the next trial, precise
LIVLNG AGK. VOL. XXVI. 1208
ly as the chances for and against the least lucky of the ten million tossers would be equal at any single tossing.
Yet, it may be urged, experience show* that luck continues; for many have won by following the lead of lucky players. Now we might at the outset, point out that this belief in the continuance of luck is suggested by an idea directly contradictory to that on which is based the theory of the maturity of the chances. If the oftener an event has occurred, the more unlikely is its occurrence at the uext trial — the common belief— then contrary to the common belief, the oftener a player has won. (that is. the longer has been his run of luck), the more unlikely is he to win at the next venture. We cannot separate the two theories, and assume that the theory of the maturity of the chances relates to the play, and the theory of runs of luck to the player. The success of the player at any trial is as distinctly an event — a chance event — as the turning up of ac« or deuce at the cast of a die.
What then are we to say of the experience of those who have won money by following a lucky player V Let us revert toour coin-tossers. Let us suppose that the progress of the venture in a given county is made known to a set of betting men in> that county; and that when it becomes known that a person has tossed "head" twelve times running, the betting men hasten to back the luck of that person. Further, suppose this to happen in every county in England. Now we have seen that these persons are no more likely to toss a thirteenth "head," than they are to fail About half will succeed, and about half will fail. Thus about half their backers will win and about half will lose. But the successes of their winners will be widely announced; while the mischances of the losers will be concealed. This will happen — the like notoriously does happen — for two reasons. First, gamblers pay little attention to the misfortunes of their fellows: the professed gambler is utterly selfish, and, moreover, he hates the sight of misfortune because it unpleasantly reminds him of his own risks. Secondly, losing gamblers do not like their losses to be noised abroad; they object to having their luck suspected by others, and they are even disposed to blind themselves to their own ill-fortune as far as possible. Thus, the inevitable success of about one half of our coin-tossers would be accompanied inevitably by the success of those who "backed their luck," and the successof such backers would be bruited abroad and be quoted as examples; while the fail