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.).