The expected value is a fundamental concept in probability theory and statistics, representing the average outcome of a random variable over numerous trials. It is calculated as the weighted average of all possible values that the variable can take, where the weights are the probabilities of each value occurring.

For discrete random variables, the expected value is computed using the formula: E[X] = Σ (xi * p(xi)), where xi represents the possible values and p(xi) their corresponding probabilities. For continuous random variables, the expected value is determined using integrals over the probability density function (PDF) of the variable ².

In practical terms, the expected value helps in decision-making processes, such as evaluating the fairness of a game or predicting outcomes in uncertain situations. For example, if you were to play a game where you win or lose money based on coin tosses, the expected value would indicate whether the game is favorable or not ¹.