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was purely accidental. We may always be tolerably sure that in a large number of tossings, about one-half will be head and about one-half tail. But when only a few tossings are to be made, this proportion can no longer be looked for with the same high degree of probability. When, again, only four or five chances are left, we may find these all dropping off at once, on the one hand, or one or two of them may run on with five or six more successful tossings; and as at each tossing the prize, already amounting for the last trial to as many pounds as there were originally chances, is doubled, we may find the average price of each chance increased by 11., 21., 4l., 81., 16/., or more, by the continued success of the longest lasting trial, or perhaps of two or three lasting equally long. This happened in the 8,192 trials whose results are recorded by De Morgan. I find that the total amount which would have been due in prizes, according to the Petersburg plan, would have been 150,830/., an average of 18/. Ss. 2 1/2d. (almost exactly) per trial; whereas the average for 8,192 trials on my plan would be only 15/.
It is manifest that, though in a million trials by this method some such sum as 301. per trial would probably cover all the prizes gained, it would be unsafe to put any definite price on each venture, where the number of venturers would of necessity be unlimited. And since even a price which would barely cover the probable expenses would be far more than speculators would care to give, the plan is utterly unsuited for a public lottery. It may be well to note how large a proportion of the speculators would lose by their yen-tare, even in a case where the total ventured was just covered by the prizes. Suppose there were rather more than a million speculators (more exactly, that the numbers were the 20th power of 2, or 1,048,576), and that the average result followed, the price per venture being 22l. Then 524,288 persons would receive only 21. and lose 20/. each; 262,144 would receive only 41. and lose 18/. each; 131,072 would receive 8/. and lose 14/. each; 65,536 would receive 16/. and lose 61. each. All the rest would gain; 32,768 would receive 32/. and gain 10/. each; 16,384 would receive 64/. and gain 42/. each; and so on; 8,192 would receive 128/. each; 4,096 would receive 256l. each; 2,048 each 512/.; 1,024 each 1,024/.; 512 each 2,048/.; 256 each 4,096/.; 138 each 8,192/.; 64 each 16,384/.; 32 each 32,768/.; 16 each 65,536/.; 8 each 131,072/.; 4 each 262,1441.; 2 each 524,288/.; I would receive 1,048,572/.; and lastly, one would receive 2,097,952/. But there would be only 65,586 out of 1,048,576 speculators who would gain, or only 1 in 16.
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