The expected gain (or expected return) is the weighted-average most likely outcome in gambling, probability theory, economics or finance.

Discrete scenarios

In gambling and probability theory, there is usually a discrete set of possible outcomes. In this case, expected gain is a measure of the relative balance of win or loss weighted by their chances of occurring.

For example, if a fair die is thrown and numbers 1 and 2 win £1, but 3-6 lose £0.5, then the expected gain per throw is

£1 × 1/3 - £0.5 × 2/3 = £0:

the game is thus fair.

Continuous scenarios

In economics and finance, it is more likley that the set of possible outcomes is continuous (a numerical or currency value between 0 and infinity). In this case, simplifying assumptions are made about the distribution of possible outcomes. Either a continuous probability function is constructed, or a discrete probability disribtuion is assumed.