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ersons who would have won every game they had played, and as many who had lost every game. These would represent the persistently lucky and unlucky men of our gambling community. There would be 625,000 who, having won four times in succession, now lost, and as many who, having lost four times in succession, now won. These would be the examples of luck--good or bad--continued to a certain stage, and then changing. The balance of our 20,000,000, amounting to seventeen millions and a half, would have had varying degrees of luck, from those who had won four games (not the first four) and lost one, to those who had lost four games (not the first four) and won but a single game. The bulk of the seventeen millions and a half would include those who would have had no reason to regard themselves as either specially lucky or
specially unlucky. But 1,250,000 of them would be regarded as examples of a change of luck, being 625,000 who had won the first three games and lost the remaining two, and as many who had lost the first three games and won the last two.
Thus, after the fifth game, there would be only 1,250,000 of those regarded (for the nonce) as persistently lucky or unlucky (as many of one class as of the other), while there would be twice as many who would be regarded by those who knew of their fortunes, and of course by themselves, as examples of change of luck, marked good or bad luck at starting, and then bad or good luck.
So the games would proceed, half of the persistently lucky up to a given game going out of that class at the next game to become examples of a change of luck, so that the number of the persistently lucky would rapidly diminish as the play continued. So would the number of the persistently unlucky continually diminish, half going out at each new encounter to join the ranks of those who had long been unlucky, but had at last experienced a change of fortune.
After the twentieth game, if we suppose constant exact halving to take place as far as possible, and then to be followed by halving as near as possible, there would be about a score who had won every game of the twenty. No amount of reasoning would persuade these players, or those who had heard of their fortunes, that they were not exceedingly lucky persons--not in the sense of being lucky because they had won, but of
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