Expected Value Calculator
Compute the expected value (mean) of a discrete probability distribution by entering up to three outcomes and their probabilities. Ideal for decision analysis, gambling odds, and statistics coursework.
About this calculator
The expected value (E[X]) is the probability-weighted average of all possible outcomes of a random variable. For a discrete distribution with outcomes x₁, x₂, x₃ and corresponding probabilities p₁, p₂, p₃, the formula is: E[X] = x₁·p₁ + x₂·p₂ + x₃·p₃. Probabilities must sum to 1 (i.e., p₁ + p₂ + p₃ = 1). The expected value does not need to equal any actual outcome — it represents the long-run average if the experiment were repeated many times. In finance, E[X] quantifies average return; in games of chance, it determines whether a bet is fair. A positive expected value means a favorable outcome on average, while a negative expected value indicates a net loss over time.
How to use
Suppose you play a game with three outcomes: win $10 with probability 0.2, win $2 with probability 0.5, and lose $5 with probability 0.3. Enter Outcome 1 = 10, Probability 1 = 0.2; Outcome 2 = 2, Probability 2 = 0.5; Outcome 3 = −5, Probability 3 = 0.3. The calculator computes: E[X] = 10×0.2 + 2×0.5 + (−5)×0.3 = 2 + 1 − 1.5 = 1.5. Your expected gain per game is $1.50.
Frequently asked questions
What does the expected value tell you about a probability distribution?
The expected value is the long-run average outcome if an experiment is repeated many times under identical conditions. It summarizes the center of a probability distribution in a single number. For example, an expected value of $1.50 in a game means that over thousands of plays, you'd average $1.50 profit per round. It does not predict what will happen in any single trial.
How do I make sure my probabilities are entered correctly in the expected value calculator?
All probabilities must be non-negative decimal values between 0 and 1, and they must sum exactly to 1. For example, use 0.25, 0.50, and 0.25 — not 25, 50, and 25. If your probabilities don't add up to 1, the result will be mathematically incorrect. Double-check by adding p₁ + p₂ + p₃ before submitting.
When should I use expected value versus median to summarize a distribution?
Use expected value when outcomes are symmetric or you care about long-run averages, such as in investment returns or insurance pricing. Use the median when your distribution is skewed or contains extreme outliers that could distort the mean. For example, household income distributions are right-skewed, making the median a more representative typical value. In repeated-game scenarios, expected value is almost always the right metric.