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ight readily calculate the probable number of losers, and would be absolutely certain that the real number could not differ greatly from that calculated; but he could not definitely assert that so many had lost, or that the total losses amounted to so much.
It occurred to the Russian Government, which has at all times been notably ready to take advantage of scientific discoveries, that a method might be devised for despoiling the public more effectually than by the Geneva method. A plan had been invented by those who wanted the public money, and mathematicians were simply asked to indicate the just price for tickets, so that the Government, by asking twice that price, or more, might make money safely and quickly. The plan turned out to be wholly impracticable; but the idea and the result of its investigation are so full of interest and instruction that I shall venture to give a full account of them here, noting that the reader who can catch the true bearing of the problem involved may consider himself quite safe from any chance of being taken in by the commoner fallacies belonging to the subject of probabilities.
The idea was this :-- Instead of the drawing of numbers, the tossing of a coin was to decide the prize
to be paid, and there were to be no blanks. If ' head' was tossed at a first trial the speculator was to receive a definite sum--2/, we take for convenience, and also because this seems to have been nearly the sum originally suggested in Russian money. If' head' did not appear till the second trial the speculator was to receive 41.; if ' head' did not appear till the third trial, he received 81.; if not till the fourth, he received 16/.; if not till the fifth, 32/.; till the sixth, 64/.; the seventh 128/.; the eighth, 256/., and so on; the prize being doubled for each additional tossing before 'head' appeared. It will be observed that the number of pounds in the-prize is 2 raised to the power corresponding to the number of that tossing at which 'head' first appears. If it appears first, for instance, at the tenth trial, then we raise 2 to the 10th power, getting 1,024, and the prize is 1,024/.; if ' head' appears first at the twelfth trial, we raise 2 to the 12th power, getting 4,048, and the prize is 4,048/.
Doubtless the origin of this idea was the observed circumstance that the more speculative ventures had a great charm for the common mind. Despite the enormous deduction made from the just value of the prize, when reties, quaternes, and other such ventures were made, the public in France, Switzerland, and Italy bought these ventures by millions, as was shown by the fact that several times in each year even quaternes were won. Now in the Petersburg plan there was a chance, however small, of enormous winnings. Head might not appear till the tenth, twelfth, or even the twentieth
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