BETTING ON HORSE RACES: OR, THE STATE OF THE ODDS.

There appears every day in the newspapers an account of the betting on the principal forthcoming races. The betting on such races as the Two Thousand Guineas, the Derby, and the Oaks, often begins more275 than a year before the races are run; and during the interval, the odds laid against the different horses engaged in them vary repeatedly, in accordance with the reported progress of the animals in their training, or with what is learned respecting the intentions of their owners. Many who do not bet themselves, find an interest in watching the varying fortunes of the horses which are held by the initiated to be leading favourites, or to fall into the second rank, or merely to have an outside chance of success. It is amusing to notice, too, how frequently the final state of the odds is falsified by the event; how some ‘rank outsider’ will run into the first place, while the leading favourites are not even ‘placed.’

It is in reality a simple matter to understand the betting on races (or contests of any kind), yet it is astonishing how seldom those who do not actually bet upon races have any inkling of the meaning of those mysterious columns which indicate the opinion of the betting world respecting the probable results of approaching contests, equine or otherwise.

Let us take a few simple cases of ‘odds,’ to begin with; and, having mastered the elements of our subject, proceed to see how cases of greater complexity are to be dealt with.

Suppose the newspapers inform us that the betting is 2 to 1 against a certain horse for such and such a race, what inference are we to deduce? To learn this let us conceive a case in which the true odds against a certain event are as 2 to 1. Suppose there276 are three balls in a bag, one being white, the others black. Then, if we draw a ball at random, it is clear that we are twice as likely to draw a black as to draw a white ball. This is technically expressed by saying that the odds are 2 to 1 against drawing a white ball; or 2 to 1 on (that is, in favour of) drawing a black ball. This being understood, it follows that, when the odds are said to be 2 to 1 against a certain horse, we are to infer that, in the opinion of those who have studied the performance of the horse, and compared it with that of the other horses engaged in the race, his chance of winning is equivalent to the chance of drawing one particular ball out of a bag of three balls.

Observe how this result is obtained: the odds are 2 to 1, and the chance of the horse is as that of drawing one ball out of a bag of three—three being the sum of the two numbers 2 and 1. This is the method followed in all such cases. Thus, if the odds against a horse are 7 to 1, we infer that the cognoscenti consider his chance equal to that of drawing one particular ball out of a bag of eight.

A similar treatment applies when the odds are not given as so many to one. Thus, if the odds against a horse are as 5 to 2, we infer that the horse’s chance is equal to that of drawing a white ball out of a bag containing five black and two white balls—or seven in all.

We must notice also that the number of balls may be increased to any extent, provided the proportion277 between the total number and the number of a specified colour remains unchanged. Thus, if the odds are 5 to 1 against a horse, his chance is assumed to be equivalent to that of drawing one white ball out of a bag containing six balls, only one of which is white; or to that of drawing a white ball out of a bag containing sixty balls, of which ten are white-and so on. This is a very important principle, as we shall now see.

Suppose there are two horses (amongst others) engaged in a race, and that the odds are 2 to 1 against one, and 4 to 1 against the other-what are the odds that one of the two horses will win the race? This case will doubtless remind my readers of an amusing sketch by Leech, called—if I remember rightly—‘Signs of the Commission.’ Three or four undergraduates are at a ‘wine,’ discussing matters equine. One propounds to his neighbour the following question: I say, Charley, if the odds are 2 to 1 against Rataplan, and 4 to 1 against Quick March, what’s the betting about the pair?’—‘Don’t know, I’m sure,’ replies Charley; ‘but I’ll give you 6 to 1 against them.’ The absurdity of the reply is, of course, very obvious; we see at once that the odds cannot be heavier against a pair of horses than against either singly. Still, there are many who would not find it easy to give a correct reply to the question. What has been said above, however, will enable us at once to determine the just odds in this or any similar case. Thus-the odds against one horse being 2 to 1, his chance of winning is equal to that of drawing one278 white ball out of a bag of three, one only of which is white. In like manner, the chance of the second horse is equal to that of drawing one white ball out of a bag of five, one only of which is white. Now we have to find a number which is a multiple of both the numbers three and five. Fifteen is such a number. The chance of the first horse, modified according to the principle explained above, is equal to that of drawing a white ball out of a bag of fifteen of which five are white. In like manner, the chance of the second is equal to that of drawing a white ball out of a bag of fifteen of which three are white. Therefore the chance that one of the two will win is equal to that of drawing a white ball out of a bag of fifteen balls of which eight (five added to three) are white. There remain seven black balls, and therefore the odds are 8 to 7 on the pair.

To impress the method of treating such cases on the mind of the reader, 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 ball 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. There279fore, 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 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 is280 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.).

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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 in282 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, the283 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.

I have spoken above of the published odds. The statements made in the daily papers commonly refer to wagers actually made, and therefore the uninitiated might suppose that everyone who tried would be able to obtain the same odds. This is not the case. The wagers which are laid between practised betting-men afford very little indication of the prices which would be forced (so to speak) upon an inexperienced bettor. Book-makers—that is, men who make a series of bets upon several or all of the horses engaged in a race—naturally seek to give less favourable terms than the known chances of the different horses engaged would suffice to warrant. As they cannot offer such terms to284 the initiated, they offer them-and in general success—fully—to the inexperienced.

It is often said that a man may so lay his wagers about a race as to make sure of gaining money whichever horse wins the race. This is not strictly the case. It is of course possible to make sure of winning if the bettor can only get persons to lay or take the odds he requires to the amount he requires. But this is precisely the problem which would remain insoluble if all bettors were equally experienced.

Suppose, for instance, that there are three horses engaged in a race with equal chances of success. It is readily shown that the odds are 2 to 1 against each. But if a bettor can get a person to take even betting against the first horse (A), a second person to do the like about the second horse (B), and a third to do the like about the third horse (C), and if all these bets are made to the same amount—say 1000l.—then, inasmuch as only one horse can win, the bettor loses 1000l. on that horse (say A), and gains the same sum on each of the two horses B and C. Thus, on the whole, he gains 1000l., the sum laid out against each horse.

If the layer of the odds had laid the true odds to the same amount on each horse, he would neither have gained nor lost. Suppose, for instance, that he laid 1000l. to 500l. against each horse, and A won; then he would have to pay 1000l. to the backer of A, and to receive 500l. from each of the backers of B and C. In like manner, a person who had backed each horse285 to the same extent would neither lose nor gain by the event. Nor would a backer or layer who had wagered different sums necessarily gain or lose by the race; he would gain or lose according to the event. This will at once be seen, on trial.

Let us next take the case of horses with unequal prospects of success—for instance, take the case of the four horses considered above, against which the odds were respectively 3 to 2, 2 to 1, 4 to 1, and 14 to 1. Here, suppose the same sum laid against each, and for convenience let this sum be 84l. (because 84 contains the numbers 3, 2, 4, and 14). The layer of the odds wagers 84l. to 56l. against the leading favourite, 84l. to 42l. against the second horse, 84l. to 21l. against the third, and 84l. to 6l. against the fourth. Whichever horse wins, the layer has to pay 84l.; but if the favourite wins, he receives only 42l. on one horse, 21l. on another, and 6l. on the third—that is 69l. in all, so that he loses 15l.; if the second horse wins, he has to receive 56l., 21l., and 6l.—or 83l. in all, so that he loses 1l.; if the third horse wins, he receives 56l., 42l., and 6l.—or 104l. in all, and thus gains 20l.; and lastly, if the fourth horse wins, he has to receive 56l., 42l., and 2ll.—or 119l. in all, so that he gains 35l. He clearly risks much less than he has a chance (however small) of gaining. It is also clear that in all such cases the worst event for the layer of the odds is, that the favourite should win. Accordingly, as professional book-makers are nearly always layers of odds, one often finds the success of a favourite spoken of in the286 papers as a ‘great blow for the book-makers,’ while the success of a rank outsider will be described as ‘a misfortune to backers.’

But there is another circumstance which tends to make the success of a favourite a blow to layers of the odds and vice versa. In the case we have supposed, the money actually pending about the four horses (that is, the sum of the amount laid for and against them) was 140l. as respects the favourite, 126l. as respects the second, 105l. as respects the third, and 90l. as respects the fourth. But as a matter of fact the amounts pending about the favourites bear always a much greater proportion than the above to the amounts pending about outsiders. It is easy to see the effect of this. Suppose, for instance, that instead of the sums 84l. to 56l., 84l. to 42l., 84l. to 21l., and 84l. to 6l., a book-maker had laid 8400l. to 5600l., 840l. to 420l., 84l. to 21l., and 14l. to 1l., respectively—then it will easily be seen that he would lose 7958l. by the success of the favourite; whereas he would gain 4782l. by the success of the second horse, 5937l. by that of the third, and 6027l. by that of the fourth. I have taken this as an extreme case; as a general rule, there is not so great a disparity as has been here assumed between the sums pending on favourites and outsiders.

Finally, it may be asked whether, in the case of horses having unequal chances, it is possible that wagers can be so proportioned (just odds being given and taken), that, as in the former case, a person backing or287 laying against all the four shall neither gain nor lose. It is so. All that is necessary is, that the sum actually pending about each horse shall be the same. Thus, in the preceding case, if the wagers 9l. to 6l., 10l. to 5l., 12l. to 3l., and 14l. to 1l., are either laid or taken by the same person, he will neither gain nor lose by the event, whatever it may be. And therefore, if unfair odds are laid or taken about all the horses, in such a manner that the amounts pending on the several horses are equal (or nearly so), the unfair bettor must win by the result. Say, for instance, that instead of the above odds, he lays 8l. to 6l., 9l. to 5l., 11l. to 3l. and 13l. to 1l., against the four horses respectively; it will be found that he must win 1l. Or if he takes the odds 18l. to 11l., 20l. to 9l., 24l. to 5l., and 28l. to 1l. (the just odds being 18l. to 12l., 20l. to 10l., 24l. to 6l., and 28l. to 2l. respectively), he will win 1l. by the race. So that, by giving or taking such odds to a sufficiently great amount, a bettor would be certain of pocketing a large sum, whatever the event of a given race might be.

In every instance, a man who bets on a race must risk his money, unless he can succeed in taking unfair advantages over those with whom he bets. My readers will conceive how small must be the chance that an unpractised bettor will gain anything but dearly-bought experience by speculating on horse-races. I would recommend those who are tempted to hold another opinion to follow the plan suggested by Thackeray in a similar case—to take a good look at professional and288 practised betting-men, and to decide ‘which of those men they are most likely to get the better of’ in turf transactions.

(From Chambers’s Journal, July 1869.)