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ving us one speculator who receives 1,024l., and still leaving one who has not yet tossed 'head.' Since each halving
gives us 1,024l., we now have last speculator tosses ' head' at the next trial and wins 2,048/.; making a grand total of twelve times 1,0241., or twelve times as many pounds as there are speculators. The average, therefore, amounts to 12/. per chance; and we see, by the way in which the result has been obtained, that in every such case the chance will be worth 2/. more than as many pounds as there are halvings. Of course the number of halvings is the number representing the power to which two is raised to give the number of speculators. The number of speculators need not necessarily be a power of 2. We have only supposed it so for simplicity of calculation. But the application of the method of halving can be almost as readily made with any number of speculators It is only when we get down to small numbers, as 9, 7, 5, or 3, that any difficulty arises from fractional or half men; but the result is not materially affected where the original number is large, by taking 4 or 3 as the next halving after either 7 or 9 (for example), or 2 as the next halving after 3. But practically we need not carry out these halvings, after we have once satisfied ourselves of the validity of the general rule. Thus, suppose we require to ascertain a fair value for a million chances. We find that the nearest power of 2 to the number one million is the 20th: 22/., then, is a fair value.
But of course, the whole train of our reasoning proves that while probably 22/. would be a fair value for a million ventures, it could not be the mathematically
that after the million ventures, another million will not be taken ? Now for two million ventures the probable value according to our method would be 23/., since two millions is nearly equal to 2 raised to the 21st power. There might be a million million ventures; and if 22/. were really the true price for one million, it would be, the true price for each of the million ventures. But since a million million are roughly equal to 2 raised to the 40th power, the price according to our method would be about 421. per chance.
All that can be said is that among any definite number of trials it is not antecedently probable that there will be any of those very long runs of ' trials' which are practically certain to occur when many times that number of trials (whatever it may be) are made.
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