In probability, a game is a scenario where a player has a chance to win rewards based on the outcome of a probabilistic event.

The rewards that a player can earn follow a probability distribution, for example X, that governs the likelihood of winning each reward.

For example, a simple game might be one where a player rolls a balanced 6-sided dice, and wins the value of their roll in dollars.

The expected return is the reward that a player can expect to earn, on average. It is given by E(X), where X is the probability distribution of the rewards.

Games can also have a cost, which is the price a player must pay each time before playing the game.

If the cost is equal to the expected return, the game is said to be fair.

Example

A player rolls a fair 6 sided dice, and wins the value of their roll in dollars. Find the cost to play, C, in dollars given that the game is fair.

The expected return is

or 3.5$.