To determine the expected return of Derby first assume that it is the same on all bets. Then imagine placing a bet of 1/x on each quinella, where x is the payoff odds on that horse, on a "for one" basis. In this case regardless of which horse won the player would get back 1 unit. The amount spent on wagers, in the above example, would be 1/5 + 1/39 + 1/200 + … + 1/79 = 1.168847. So as in any bet the expected return is the ratio of the amount the player can expect to get back to the amount bet. In this case 1/1.168847 = 0.855544 = 85.54%. The house edge is thus 1 - 0.855544 = 0.144456 = 14.46%.

 Inverse of win: How much the player must bet for a return of 1.

Fair probability: The actual probability of winning.

Fair odds: What the bet should bet to have a 100% return.

The expected return is the inverse of the total "inverse of win" column. In this case 1/1.168847 = 85.54%.

Next let’s look at a futures bet. Unlike Derby sports futures pay on a "to one" basis. I think it is easier to understand them if you convert the odds to a "for one" basis, by adding 1 to each win. Then take the inverse of each win. The sum of these inverses is how much you have to bet to be guaranteed a return of 1. So in the example below the expected return is 1/1.3336 = 74.99%, i.e. the house edge is 25.01%. The "fair probability" column is the actual probability of winning, assuming each bet had the same house advantage. Both "fair odds" columns are what the bet should be to have a 100% expected return.

2005 National League Championship

Source: Bodog