POSTSCRIPT

WHILST this book is still in the press, an article on 'Science and Monte Carlo,' by Professor Karl Pearson, has appeared in the (monthly) 'Fortnightly Review.' This article deals with the game of roulette, and is one which may be commended to the perusal of all who may have any pet theories in connection with chance and luck. It constitutes, in fact, a very serious impeachment of the validity of all accepted theories of chance; so serious, indeed, that one stands amazed at the discrepancies which are revealed, and their having remained so long unnoticed. There appears to be no way out of the difficulty. Either roulette is not a game of chance, or the doctrines of chance are utterly wrong.

It appears from Professor Pearson's investigations, that in a given number of throws the results shown by the "even-money" chances are fairly in accord with the theory as a whole. That is to say, the odd and even numbers, the red and black, turn up respectively in very nearly equal proportions. Also the 'runs' or sequences of odd or even are such as would not give rise to any conflict between theory and practice. But the astounding fact is that the 'runs' or successions of red or black occur in a manner which is utterly at variance with theory. Why this should be so, and why 'red and black' should thus prove to be an exception to the theory, whilst 'odd and even' is not, passes the wit of man to comprehend.

In one of the cases quoted by Professor Pearson, 8,178 throws of the roulette-ball are compared with a similar number of tosses of a coin, and both results are checked against the theoretical probabilities. In tossing a coin or throwing a roulette-ball 8,178 times, theory demands that the number of throws which do not result in sequences that is to say, throws in which head is followed by tail, or red by black should be 2,044. Those are the probabilities of the case. But the actual results were as follows:

Theory . . . . . . . . 2,044

Roulette . . . . . . . 2,462

Tossing . . . . . . . 2,168

There are too many single throws in each case, but the results given by tossing were much nearer the theoretical proportion than in the case of the roulette. Proceeding a step further, we find that the sequences of two work out thus:

Theory . . . . . . . . 1,022

Roulette . . . . . . . . 945

Tossing . . . . . . . 1,056

Here the figures given by roulette are far too small. This is found to be the case with sequences of three and four also. When we come to sequences of five, however, the numbers stand:

Theory . . . . . . . . . 128

Roulette . . . . . . . . 135

Tossing . . . . . . . . 120

In this case, the roulette is nearer the mark than the tossing; and from this point onward through the higher sequences, roulette gives numbers which are far too high. For instance, in sequences of eight, theory says that there should be 16, but roulette gives 30. In sequences of eleven theory says 2, but roulette gives 5. Arriving at sequences of twelve, the figures are:

Theory . . . . . . . . . . 1

Roulette . . . . . . . . . 1

Tossing . . . . . . . . . 1

Here all the results are in accord.

This is only one instance out of several recorded by Professor Pearson; in every case the results being similar. That only one instance of such abnormal variation should occur is, theoretically, well nigh impossible; but that there should be three or four such cases in the course of a single twelvemonth is nothing short of miraculous. The chances against the occurrence of such events are enormous; and yet every case investigated shows the same kind of result. Truly this must be another example of the malignity of matter.

The practical outcome of these investigations is to emphasize the utter futility of any scheme of winning at roulette based upon the law of averages or the doctrines of chance. It is more than likely, in my opinion, that further analysis of the records of Monte Carlo would reveal similar discrepancies in other departments of the game.

Personally, I fail to see how the devotees of the 'Higher Statistics' will contrive to meet the difficulty here presented. Why roulette should obey the laws of chance in some respects and not in others, is incomprehensible from any point of view whatever. One is driven to the conclusion that human experience and human statistics are upon too limited a scale to form a sufficient basis upon which to found either the proof or disproof of any universal theory. The only refuge appears to be that, given eternity, all events, however improbable, are possible.

It is to be hoped that Professor Pearson will find an opportunity of continuing his researches in this direction, for the subject promises to be one of exceeding interest. Of course, it may be objected that the few instances given are insufficient to affect the theory materially; but, as the Professor says of one of his instances, had roulette been played constantly on this earth, from the earliest geological times to the present day, such an event might be expected to happen only once. Those who believe that an infinite number of bets, where the chances are fair and equal, can result in neither loss nor gain, should ponder this carefully. If the doctrines of chance can fail in one case, they can fail in others. At best, they are but a broken reed, and those who trust to them should beware the risk that is thereby entailed. Above all, the punter should bear in mind that, whatever theory may say or practice apparently demonstrate, the fact that any given event has happened so many times in succession makes not the slightest particle of difference to its chances of happening again, If one tossed a coin a hundred times, and it turned up 'head' every time, that would not in any way lessen its chance of turning up the same way at the next throw. The figures given in the article above referred to are neither more nor less than an illustration of this very palpable truth, extraordinary as they undoubtedly are when viewed in the light of theory.

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