Expected value

Expected value is the average outcome of a random variable, weighted by how likely each outcome is. For a discrete random variable X, the formula is E(X) = Σ x · P(x): multiply each possible value by its probability and add the products. The result is the long-run average — the value the sample mean approaches as the number of trials grows.

A quick example: a raffle ticket pays $100 with probability 0.01 and nothing otherwise. Its expected value is (100)(0.01) + (0)(0.99) = $1. If the ticket costs $2, the expected net outcome is −$1 per ticket, so a buyer loses a dollar on average over many plays. Note that the expected value need not be a possible outcome — no single ticket ever pays exactly $1.

Expected value underlies much of probability and statistics: it is the mean of a probability distribution, the basis for measures of spread like variance, and the standard tool for comparing uncertain alternatives. In business, decision makers compare projects by multiplying each scenario's payoff by its probability; in insurance and finance, expected loss and expected return drive pricing.

Expected value appears on a wide range of exams. The ACT tests it in probability word problems, the FE Mechanical exam covers it within probability and statistics alongside dispersion and common distributions, and the CMA Part 1 exam applies it as a forecasting technique for weighting scenarios in planning and budgeting.