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ibed in the preceding paragraph as the average outcome of 1,048,576 ventures. In other words, no one puts so much faith in his luck as to venture a sum on the chance of gaining a little if he tosses
tail' four times running (losing if ' head' appears sooner), and of gaining more and more the oftener 'tail' is tossed, until, should he toss tail 20 times running, he will receive more than two million pounds. But almost every person who is willing to gamble at all will be ready to venture the same sum on the practically equivalent chance of winning in a lottery where there are rather more than a million tickets, and the same prizes as in the other case. Whatever advantage there is, speaking mathematically, is in favor of the tossing risk; for the purchaser of a trial has not only the chance of winning such prizes as in a common lottery arranged to give prizes corresponding to the above-described average case, but he has a chance, though a small one, of winning four, eight, sixteen, or more millions of pounds for his venture of 22l. We see then that the gamblers are very poor judges of chances, rejecting absolutely risks of one kind, while accepting systematically those of another kind, though of equal mathematical value, or even greater.
In passing, I may note that the possibility of win-
ning abnormally valuable prizes in the Petersburg lottery affords another explanation of the apparent paradox involved in the assertion that no sum, however larger fairly represents the mathematical value of each trial. To obtain the just price of a lottery-ticket, we must multiply each prize by the chance of getting it, and add the results together; this is the mathematical value of one chance or ticket. Now in the Petersburg lottery the possible prizes are 21., 41., 81., 16/., and so on, doubling to infinity; the chances of getting each are, respectively, one-half, one-fourth, one-eighth, one-sixteenth, and so on. The value of a chance, then, is the half of 2l. added to the quarter of 4/., to the eighth of 8/., and so on to infinity, each term of the infinite series being 1/. Hence the mathematical value of a single chance is infinite. The result appears paradoxical; but it really means only that the oftener the trial is made, the greater will be the probable average value of the prizes obtained. Or, as in fact the solution is that if the number of trials were infinite the value of each would be infinite, we only obtain a paradoxical result in an impossible case. Note also that the two kinds of infinity involved in the number of trials and in the just mathematical price of each are different. If the number of trials were 2 raised to an infinitely high power, the probable average value of each trial would be the infinitely high number representing that power. But 2 raised to that power would give an infinitely higher number. To take very large numbers instead of infinite numbers, which simply elude us :--Suppose the number
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