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Expected value (E(X)) is the long-run average outcome of a random variable. It is the probability-weighted sum of all possible values: E(X) = Σ xᵢ · P(xᵢ).
Think of it as: if you repeated an experiment thousands of times, E(X) is what the average outcome would converge to. A fair six-sided die has E(X) = (1+2+3+4+5+6)/6 = 3.5 — you never roll 3.5, but over many rolls your average approaches 3.5.
Expected value is fundamental to decision theory, insurance pricing, gambling analysis, and finance. A positive expected value means a bet or investment favors you on average; negative means it favors the house or counterparty.
Worked example — a game pays $10 if you flip heads and costs $3 if you flip tails (fair coin). E(X) = 10 × 0.5 + (−3) × 0.5 = 5 − 1.5 = $3.50 per play. This game has a positive expected value of $3.50.
E(X) = Σ xᵢ · P(xᵢ)
Probabilities must sum to 1.00
| Outcome (x) | Probability P(x) | |
|---|---|---|
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.