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The expected value is the long-run average of a random variable's outcomes as the sample size approaches infinity. It represents the mean value you would expect over many repeated trials. Calculated by multiplying each possible outcome by its probability and summing these products, the expected value provides a single number summarizing the central tendency of a probability distribution.

As the number of trials increases, the sample mean fluctuates less and gradually approaches a constant value. This convergence demonstrates that with more data, the observed average becomes increasingly stable and reliable. The expected value represents this limiting mean value that emerges when sample size grows infinitely large, illustrating the law of large numbers in action.

Expected value is calculated by summing the products of each event and its probability: E(X) = Σ[x · P(x)], where x represents each possible outcome and P(x) is its probability. This formula mirrors the standard mean calculation but weights each outcome by how likely it is to occur, providing a probability-adjusted average.

Expected value quantifies the average outcome of repeated bets, revealing whether a wager favors the player or house. In roulette, betting ten dollars on a single number yields an expected value of negative 53 cents per bet, meaning you lose money on average. This calculation helps decision-makers evaluate risk and determine whether a gamble is worth taking long-term.

Expected value provides a rational framework for comparing uncertain outcomes by calculating the average result of repeated decisions. It transforms subjective uncertainty into a single numerical metric, enabling informed choices about risky situations. By quantifying what you can expect to gain or lose on average, expected value guides optimal decision-making in business, finance, and personal planning.

The expected value is represented by the Greek letter μ (mu), which also denotes the population mean. This symbol emphasizes that expected value is the theoretical long-term average of a probability distribution. Using μ standardizes notation across statistics, making it clear that expected value and population mean are equivalent concepts.

A probability distribution describes all possible outcomes and their likelihoods, while expected value summarizes that distribution into a single average value. Expected value is calculated directly from the probability distribution by weighting each outcome by its probability. Understanding probability distributions is essential for computing meaningful expected values that accurately reflect the underlying random process.