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 1 wanted. Now, the odds of 10 to 1 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 1 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 little 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 favourites, 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 downward progress of the same horse in the betting. In fact, there are yet more delicate gradations, marked by such expressions respecting certain odds, as—offered freely, offered, offered and taken (meaning that some offers only have been accepted), taken, taken and wanted, wanted, and so on.
As an illustration of some of the principles I have been considering, let us take from the day’s paper,[18] the state of the odds respecting the ‘Two Thousand Guineas.’ It is presented in the following form:—
TWO THOUSAND GUINEAS.
| 7 to | 2 against | Rosicrucian (off.). |
| 6 to | 1 against | Pace (off.; 7 to 1 w.). |
| 10 to | 1 against | Green Sleeve (off.). |
| 100 to | 7 against | Blue Gown (off.). |
| 180 to | 80 against | Sir J. Hawley’s lot (t.). |
This table is interpreted thus: bettors are willing to lay the same odds against Rosicrucian as would be the true mathematical odds against drawing a white ball out of a bag containing two white and seven black balls; but no one is willing to back the horse at this rate; on the other hand, higher odds are not offered against him. Hence it is presumable that his chance is somewhat less than that above indicated. Again, bettors are willing to lay the same odds against Pace as might fairly be laid against drawing one white ball out of a bag of seven, one only of which is white; but backers of the horse consider that they ought to get the same odds as might be fairly laid against drawing the white ball when an additional black ball had been put into the bag. As respects Green Sleeve and Blue Gown, bettors are willing to lay the odds which there would be, respectively, against drawing a white ball out of a bag containing—(1) eleven balls, one only of which is white, and (2) one hundred and seven balls, seven only of which are white. Now, the three horses, Rosicrucian, Green Sleeve, and Blue Gown, all belong to Sir Joseph Hawley, so that the odds about the three are referred to in the last statement of the list just given. And since none of the offers against the three horses have been taken, we may expect the odds actually taken about ‘Sir Joseph Hawley’s lot’ to be more favourable than those obtained by summing up the three former in the manner we have already examined. It will be found that the resulting odds (offered) against Sir J. Hawley’s lot—estimated in this way—should be, as nearly as possible, 132 to 80. We find, however, that the odds taken are 180 to 80. Hence, we learn that the offers against some or all of the three horses are considerably short of what backers require; or else that some person has been induced to offer far heavier odds against Sir J. Hawley’s lot than are justified by the fair odds against his horses, severally.
I have heard it asked why a horse is said to be a favourite, though the odds may be against him. This is very easily explained. Let us take as an illustration the case of a race in which four horses are engaged to run. If all these horses had an equal chance of winning, it is very clear that the case would correspond to that of a bag containing four balls of different colours; since, in this case, we should have an equal chance of drawing a ball of any assigned colour. Now, the odds against drawing a particular ball would clearly be 3 to 1. This, then, should be the betting against each of the three horses. If any one of the horses has less odds offered against him, he is a favourite. There may be more than one of the four horses thus distinguished; and, in that case, the horse against which the least odds are offered is the first favourite. Let us suppose there are two favourites, and that the odds against the leading favourite are 3 to 2, those against the other 2 to 1, and those against the best non-favourite 4 to 1; and let us compare the chances of the four horses. I have not named any odds against the fourth, because, if the odds against all the horses but one are given, the just odds against that one are determinable, as we shall see immediately. The chance of the leading favourite corresponds to the chance of drawing a ball out of a bag in which are three black and two white balls, five in all; that of the next to the chance of drawing a ball out of a bag in which are two black and one white ball, three in all; that of the third, to the chance of drawing a ball out of a bag in which are four black balls and one white one, five in all. We take, then, the least number containing both five and three—that is, fifteen; and then the number of white balls, corresponding to the chances of the three horses, are respectively six, five, and three, or fourteen in all; leaving only one to represent the chance of the fourth horse (against which the odds are therefore 14 to 1). Hence the chances of the four horses are respectively as the numbers six, five, three and one.