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for each of these chances is clearly three times as good as the chance of drawing, at a single trial, one particular ticket out of ten.
It will be found that we can now test any wager, not merely determining whether it is fair or unfair, but the extent to which it is so, if only the actual chance of the horse or horses concerned is supposed to be known. Unfortunately, in the great majority of cases bets are unfair in another way than that which we are for the moment considering, the odds not only differing from those fairly representing the chances of the horse or horses concerned, but one party to the wager having better knowledge than the other what those chances are. Cases of this kind will be considered further on.
Suppose that the just odds against a horse in a race are 9 to 1. By this I mean that so far as the two bettors are concerned (that is, from all that they know about the chances of the horse), it is nine times more likely that the horse will not win the race than that he will. Now, it is nine times more likely that a particular ticket among ten will not be drawn at a single trial
than that it will. So the chance of this horse is correctly represented by the chance of the prize ticket being drawn in a lottery where there are ten tickets in all. If two persons arrange such a lottery, and A pays in 11. to the pool, while the other, B, pays in 91., making 10/. in all, A gets a fair return for his money in a single drawing, one ticket out of the ten being marked for the prize. A represents, then, the backer of the horse who risks 11.; B the layer of the odds who risks 91. The sum of the stakes is the prize, or 10/. If A risks less than 11., while B risks 9/., the total prize is diminished; or if, while A risks 11., B risks less than 91., the total is diminished. In either case the wrong done to the other bettor amounts precisely to the amount by which the total is diminished. If, for instance, A only wagered 18s. against B's 91., the case is exactly the same as though A and B having severally contributed l l. and 9/. to a pool, one ticket out of ten having been marked and A to have one chance only of drawing it (which we have just seen would be strictly fair), A abstracted two shillings from the pool. If B only wagered 71. instead of 9l. against A's 11. the case would be just the same as though, after the pool had been made up as just de- scribed, B had abstracted 21.
Take another case. The odds are 7 to 3 against a horse. The chance of its winning is the same as that of drawing a marked ticket out of a bag containing ten, when three are marked and seven are unmarked. We know that in this case two players, A and B, forming
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