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r, let us take the betting about three horses--say 3 to 1, 7 to 2, and 9 to 1 against the three horses respectively. Then their respective chances are equal to the chance of drawing (1) one white bah out of four, one only of which is white; (2) a white ball out of nine, of which two only are white; and (3) one white ball out of ten, one only of which is white. The least number which contains four, nine, and ten is 180; and the above chances, modified according to the principle explained above, become equal to the chance of drawing a white ball out of a bag containing 180 balls, when 45, 40, and 18 (respectively) are white. Therefore, the chance that one of the three will win is equal to that of drawing a white ball out of a bag containing 180 balls, of which 103 (the sum of 45, 40, and 18) are white. Therefore, the odds are 103 to 77 on the three.
One does not hear in practice of such odds as 103 to 77. But betting-men (whether or not they apply just principles of computation to such questions is unknown to me) manage to run very near the truth. For instance, in such a case as the above, the odds on the three would probably be given as 4 to 3--that is, instead of 103 to 77 (or 412 to 308), the published odds would be equivalent to 412 to 309.
And here a certain nicety in betting has to be mentioned. In running the eye down the list of odds, one will often meet such expressions as 10 to 1 against such a horse offered, or 10 to I wanted. Now, the odds of 10 to I taken may be understood to imply that the horse's chance is equivalent to that of drawing a certain ball out of a bag of eleven. But if the odds are offered and not taken, we cannot infer this. The offering of the odds implies that the horse's chance is not better than that above mentioned, but the fact that they are not taken implies that the horse's chance is not so good. If no higher odds are offered against the horse, we may infer that his chance is very little worse than that mentioned above. Similarly, if the odds of 10 to I are asked for, we infer that the horse's chance is not worse than that of drawing one ball out of eleven; if the odds are not obtained, we infer that his chance is better; and if no lower odds are asked for, we infer that his chance is very tittle better.
Thus, there might be three horses (A, B, and C) against whom the nominal odds were 10 to 1, and yet these horses might not be equally good favorites, because the odds might not be taken, or might be asked for in vain. We might accordingly find three such horses arranged thus:
Odds.
A , . 10 to 1 (wanted).
B 10 to 1 (taken).
C . . 10 to 1 (offered).
Or these different stages might mark the upward or
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