Expected Value Calculator
The expected value calculator computes the weighted average of possible outcomes given their probabilities. The formula is E(X) = Sum(x_i * P(x_i)). For example, if you roll a fair die, the possible outcomes are 1-6, each with probability 1/6. The expected value is (1+2+3+4+5+6) / 6 = 3.5. This calculator also computes the variance, which measures how much the outcomes vary around the expected value. Enter outcome values (comma-separated) and their corresponding probabilities; probabilities must sum to 1.
Formulas
E(X) = Sum(x_i * P(x_i))
Var(X) = E(X^2) - [E(X)]^2
StDev = sqrt(Var(X))
Distribution table
Expected value calculator: frequently asked questions
What is expected value?
Expected value (or mean) is the average outcome of a random variable weighted by probability. The formula is E(X) = Sum(x_i * P(x_i)), where x_i are the possible values and P(x_i) are their probabilities. It represents the long-term average if you repeated the experiment many times.
How do probabilities add up?
The sum of all probabilities must equal 1 (or 100 percent). This represents certainty: one of the outcomes must occur. If your probabilities do not sum to 1, the calculator will show an error.
What is variance?
Variance measures the spread of possible outcomes around the expected value. A high variance means outcomes vary widely. A low variance means outcomes are close to the expected value. Variance = E(X^2) - [E(X)]^2.
When is expected value used?
Expected value is used in games of chance, investment decisions, insurance calculations, and any decision involving risk. For example, a casino uses expected value to ensure profits over time.
Can expected value be negative?
Yes. If most outcomes are negative (like losses in a game), the expected value will be negative. The expected value reflects the average outcome, whether positive or negative.
Official sources
- Wolfram MathWorld: Expected Value.
- Wikipedia: Expected Value.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.