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 ².

In practical terms, the expected value helps in decision-making processes, such as determining whether to engage in a game or investment. For example, if you toss a coin and win $1 for heads and lose $2 for tails, the expected value of this game would be negative, indicating that it is not a favorable game to play ¹.

Understanding expected value is crucial in various fields, including finance, insurance, and machine learning, as it underpins concepts like entropy and mean squared error ¹.