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ar below the truth--there are then one million ventures each year. It cannot be regarded as wonderful, then, that among the fifty millions of ventures made (on this supposition) during the last half century, there should be noted some runs of luck which on any single trial would seem incredible. On the contrary, this is so far from being wonderful that it would be far more wonderful if no such runs of luck had occurred, It is probable that if the actual number of ventures, and the circumstances of each, could be ascertained, and if any mathematician could deal with the tremendous array of figures in such sort as to deduce the exact mathematical chance of the occurrence of bank-breaking runs of luck, it would be found that the antecedent odds were many millions to one in favor of the occurrence of a certain number of such events. In the simpler case of our coin-tossers the chance of twenty successive 'heads' being tossed can be quite readily calculated. I have made the calculation, and I find that if the ten million persons had each two trials the odds would be more than 10,000 to 1 in favor of the occurrence of twenty successive
'heads' once at least; and only a million and a half need have a single trial each, in order to give an even chance of such an occurrence.
But we may learn a further lesson from our illustrative tossers. We have seen that granted only a sufficient number of trials, runs of luck are practically certain to occur: but we may also infer that no run of luck can be trusted to continue. The very principle which has led us to the conclusion that several of our tossers would throw twenty ' heads' successively, leads also to the conclusion that one who has tossed ' heads' twelve or thirteen times, or any other considerable number of times in succession, is not more (or less) likely to toss 'head' on the next trial than at the beginning. About half, we said, in discussing the fortunes of the tossers, would toss 'head' at the next trial: in other words, about half would fail to toss 'head.' The chances for and against these lucky tossers are equal at the next trial, precisely as the chances for and against the least lucky of the ten million tossers would be equal at any single tossing. Yet, it may be urged, experience shows that luck · continues; for many have won by following the lead of lucky players. Now I might, at the outset, point out that this belief in the continuance of luck is suggested by an idea directly contradictory to that on which is based the theory of the 'maturity of the chances.' If the oftener an event has occurred, the more unlikely is its occurrence at the next trial--
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