Horse-racing results depend on many factors: the form of the competing horses, the effectiveness of their trainers, the quality of the jockeys, the length of the course, the state of the ground and the type of race (trotting or galloping)… Nevertheless, under French law, betting on horse races is classed as gambling. Which horses win can indeed be regarded as a matter of chance on which money is wagered: the "glorious uncertainty of sport" presumably owes much to chance, and betting on horses is probably no more unreasonable than buying a lottery ticket…
No probabilities, just odds!
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Those who take bets, or bookmakers, assess the likelihood of the events on which bets are placed. They could therefore use probability, a concept devised by mathematicians for precisely this purpose… but they avoid it because this measure of likelihood is additive (see box)!
To explain this, consider a race involving four horses, A, B, C and D. In the simplest case, bookmakers must assess each horse's chances of winning. In other words, they estimate the four probabilities P(A), P(B), P(C) and P(D) that A, B, C or D, respectively, will cross the finish line first. These events cover every possibility and are mutually exclusive (a photo finish rules out any possible dead heat), so probability theory requires the following equality:
P(A) + P(B) + P(C) + P(D) = 1.
