A bookmaker is taking bets on a two-horse race. Choosing to be scientific, he studies the form of both horses over various distances and goings as well as considering such factors as training, diet and choice of jockey. Eventually he correctly calculates that one horse has a 25% chance of winning, and the other a 75% chance. Accordingly the odds are set at 3—1 against and 3—1 on respectively.

But there is a degree of popular sentiment reflected in the bets made, adding up to $5000 for the first and $10000 for the second. Were the second horse to win, the bookmaker would make a net profit of $1667, but if the first wins he suffers a loss of $5000. The expected value of his profit is 25% × (−$5000) + 75% × ($1667) = $0, or exactly even. In the long term, over a number of similar but independent races, the law of averages would allow the bookmaker to break even. Until the long term comes, there is a chance of making a large loss.

Suppose however that he had set odds according to the money wagered — that is, not 3—1 but 2—1 against and 2—1 on respectively. Whichever horse wins, the bookmaker exactly breaks even. The outcome is irrelevant.

In practice the bookmaker sells more than 100% of the race and the odds are shortened to allow for profit (see table).