Expected Value Calculator

The expected value is the long-run average outcome of a random variable: each outcome weighted by its probability. Enter outcomes and probabilities to get E(X), the variance and standard deviation, with every product shown.

Outcome xProbability P(x)Remove
Expected value E(X)
20.5
Variance
1,984.75
Standard deviation
44.5505
Σ P(x)
1

Step-by-step working

  1. Multiply each outcome by its probability0 × 0.5 = 0 10 × 0.3 = 3 50 × 0.15 = 7.5 200 × 0.05 = 10
  2. Add the products: E(X) = Σ x·P(x)E(X) = 20.5
  3. For the spread, find E(X²) = Σ x²·P(x), then Var(X) = E(X²) − [E(X)]²Var(X) = 2405 − 20.5² = 1984.75 σ = √Var(X) = 44.550533

Formula E(X) = Σ x · P(x) Var(X) = Σ x² · P(x) − [E(X)]²

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How to calculate expected value

  1. List every possible outcome x and its probability P(x). The probabilities must add up to 1.
  2. Multiply each outcome by its probability.
  3. Add the products.

Example: a raffle ticket costs $5. There is a 1 in 1,000 chance of winning $2,000. Net outcomes are +$1,995 (P = 0.001) and −$5 (P = 0.999), so E(X) = 1,995 × 0.001 − 5 × 0.999 = 1.995 − 4.995 = −$3.00 per ticket.

What expected value does (and doesn’t) mean

E(X) is an average over many repetitions, not a prediction for one. A fair die has E(X) = 3.5, which you can never roll. A negative expected value for a bet means you lose on average in the long run, even though individual plays can win. For repeated yes/no trials, the binomial calculator gives E(X) = np directly.

Frequently asked questions

What is the expected value of rolling a die?

3.5: (1 + 2 + 3 + 4 + 5 + 6) × 1/6 = 21 ÷ 6.

Can expected value be negative?

Yes, whenever outcomes are losses on average, as with most lottery and casino games.

How is expected value different from a weighted average?

It is a weighted average whose weights are probabilities that add up to 1.