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 there 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 proportion 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.