Expected value calculator
Expected value is the long-run average: each outcome weighted by how often it happens. See which outcomes pull the balance point.
Probabilities add up to 1
Weights are sized by probability. The orange pivot sits where the beam balances: the expected value.
| x | P(x) | x · P(x) | (x − μ)² · P(x) |
|---|---|---|---|
| 0 | 0.5 | 0 | 210.125 |
| 10 | 0.3 | 3 | 33.075 |
| 50 | 0.15 | 7.5 | 130.5375 |
| 200 | 0.05 | 10 | 1611.0125 |
| Σ | 1 | 20.5 | 1984.75 |
Expected value is a balance point
Put a weight at each outcome, sized by its probability, on a weightless beam. The point where the beam balances is the expected value, E(X) = Σ x·P(x). It is the average you would get per trial over a very long run, even if it is not a value you can get on any single trial.
Worked example: a $5 raffle
1,000 tickets, one $500 prize and ten $50 prizes. Net of the $5 ticket, you win $495 with probability 0.001, $45 with probability 0.01, and lose $5 otherwise. E(X) = 495(0.001) + 45(0.01) − 5(0.989) = −$4.00. On average each ticket costs you $4; the organiser keeps $4,000 of every $5,000 raised.
Why variance matters too
Two bets can have the same expected value but very different risk. Variance, Σ(x − μ)²·P(x), measures how widely results swing around the average. Insurance works because customers accept a negative expected value in exchange for much lower variance.
Watch a running average close in on the expected value with the law of large numbers simulator.
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Questions students ask
How do you calculate expected value?
Multiply each outcome by its probability and add the products: E(X) = Σ x·P(x). The probabilities must add up to 1.
How do you find the variance of a discrete random variable?
Var(X) = Σ (x − E(X))² · P(x), or equivalently E(X²) − E(X)². The standard deviation is its square root.
What does a negative expected value mean?
For a bet, it means you lose money on average over many plays. A US roulette bet on red has an expected value of about −5.3 cents per dollar.
Do you ever actually get the expected value?
Often not. The expected value of a die roll is 3.5, which you can never roll. It is the average you approach over many repetitions (the law of large numbers).