The expected gain (or expected return) is the weighted-average most likely outcome in gambling, probability theory, economics or finance. Gambling (or betting) is any behavior involving risking money or valuables (making a wager or placing a stake) on the outcome of a game, contest, or other event in which the outcome of that activity depends partially or totally upon chance or upon ones ability to do something. ... Probability theory is the mathematical study of probability. ... Economics (deriving from the Greek words οίκω [oeko], house, and νέμω [nemo], distribute) is the social science that studies the allocation of scarce resources. ... Finance studies and addresses the ways in which individuals, businesses and organizations raise, allocate and use monetary resources over time, taking into account the risks entailed in their projects. ...

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. Gambling (or betting) is any behavior involving risking money or valuables (making a wager or placing a stake) on the outcome of a game, contest, or other event in which the outcome of that activity depends partially or totally upon chance or upon ones ability to do something. ... Probability theory is the mathematical study of probability. ...

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 likely 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 distribution is assumed. Economics (deriving from the Greek words οίκω [oeko], house, and νέμω [nemo], distribute) is the social science that studies the allocation of scarce resources. ... Finance studies and addresses the ways in which individuals, businesses and organizations raise, allocate and use monetary resources over time, taking into account the risks entailed in their projects. ...