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emove from the question the perplexities resulting from the nature of the above-named games, let us suppose that the tossing of a coin is to determine the success Or failure of the player, and that he will win if he throws ' head.' Now if a player tossed, head' twenty times running on any occasion it would be regarded as a most remarkable run of luck, and it ·would not be easy to persuade those who witnessed the! occurrence that the thrower was not in some special and definite manner the favorite of Fortune. We may take such exceptional success as corresponding to the good fortune of a 'bank-breaker.' Yet it is easily shown that with a number of trials which must fall enormously short of the number of cases in which fortune is risked at foreign Knrsaals, the throwing of twenty successive ' heads' would be practically insured. Suppose every adult person in Britain--say 10,000,000 persons in all--were to toss a coin, each tossing until ' tail' was thrown; then it is practically certain that
several among them would toss twenty times before ' tail' was thrown. Thus: It is certain that about five millions would toss, head' once; of these about one- half, or some two millions and a half, would toss ' head'"! on the second trial; about a million and a quarter would toss ' head' on the third trial; about six hundred thousand on the fourth; some three hundred thousand on the fifth; and by proceeding in this way--roughly halving the numbers successively obtained--we find that some eight or nine of the ten million persons would be almost certain to toss 'head' twenty times running. It must be remembered that so long as the numbers continue large the probability that about half will toss ' head' at the next trial amounts almost to Certainty. For example, about 140 toss 'head' sixteen times running: now, it is utterly unlikely that of these 140, fewer than sixty will toss 'head' yet a seventeenth time. But if the above process failed on trial to give even one person who tossed 'heads' twenty times running--an utterly improbable event-- yet the trial could be made four or five times, with practical certainty that not one or two, but thirty or forty, persons would achieve the seemingly incredible feat of tossing ' head' twenty times running. Nor would all these thirty or forty persons fail to throw even three or four more ' heads.' Now, if we consider the immense number of trials made at gambling-tables, and if we further consider the gamblers as in a sense typified by our ten millions of coin tossers, we shall see that it is not merely
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