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e following simple illustrative case :-
Suppose a large number of persons--say, for instance, twenty millions--engage in some game depending wholly on chance, two persons taking part in each game, so that there are ten million contests. Now, it is obvious that, whether the chances in each contest are exactly equal or not, exactly ten millions of the twenty millions of persons will rise up winners and as many will rise up losers, the game being understood to be of such a kind that one player or the other must win. So
far, then, as the results of that first set of contests are concerned, there will be ten million persons who will consider themselves to be in luck.
Now, let the same twenty millions of persons engage a second time in the same two-handed game, the pairs of players being not the same as at the first encounter, but distributed as chance may direct. Then there will be ten millions of winners and ten millions of losers. Again, if we consider the fortunes of the ten million winners on the first night, we see that, since the chance which each one of these has of being again a winner is equal to the chance he has of losing, about one-half of the winning ten millions of the first night will be winners on the second night too. Nor shall we deduce a wrong general result if, for convenience, we say exactly one-half; so long as we are dealing with very large numbers we know that this result must be near the truth, and in chance problems of this sort we require (and can expect) no more. On this assumption, there are at the end of the second contest five millions who have won in both encounters, and five millions who have won in the first and lost in the second. The other ten millions, who lost in the first encounter, may similarly be divided into five millions who lost also in the second, and as many who won in the second. Thus, at the end of the second encounter, there are five millions of players who deem themselves lucky, as they have won twice and not lost at all; as many who deem themselves unlucky, having lost in both encounters; while ten millions, or half the original number, have no reason
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