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 |