How to calculate expected value
Expected value is the probability-weighted average of all outcomes: E(X) = Σ(x · p). Multiply each outcome by its probability, then add the products. For outcomes 10, 20, 30, 40 with probabilities 0.1, 0.2, 0.3, 0.4 the products are 1, 4, 9, 16, so E(X) = 30.
There is no dividing step, because the probabilities supply the weights and already total 1. That is why E(X) always falls between the smallest and largest outcome. Each outcome needs a probability that you can work out with the probability calculator before the weighting makes sense.
How to use this calculator
Enter each outcome with its probability, leaving unused pairs blank. The badge shows the sum of your probabilities and warns if they don't add up to about 1. For a single average of plain numbers, use the median calculator instead.
Enter probabilities as decimals, not percentages: 30% is 0.3, not 30. Outcomes can be money, points or minutes, and E(X) comes back in that same unit.
Probabilities must sum to 1
For a valid probability distribution the probabilities should sum to 1 (100%). If they don't, recheck your figures — the expected value will be off. Expected value underlies risk and gambling math, and pairs naturally with spread measures like the variance calculator.
A sum below 1 usually means an outcome was left out; a sum above 1 means two outcomes overlap. Expected value also says nothing about risk: a bet with E(X) = 30 and a certain 30 look identical here.
Frequently asked questions
- How do I calculate expected value?
- Multiply each outcome by its probability and sum the products: E(X) = Σ(x·p). For 10, 20, 30, 40 at 0.1, 0.2, 0.3, 0.4 the expected value is 30.
- Do the probabilities have to sum to 1?
- Yes, for a valid distribution. The calculator shows the sum and warns if it isn't close to 1.
- What does expected value mean?
- It is the long-run average outcome you'd expect if the situation were repeated many times.
- Can outcomes be negative?
- Yes. Negative outcomes (like losses) are allowed and lower the expected value.
- Why is the expected value sometimes impossible to actually get?
- Because it is an average, not a prediction of any single trial. A fair die has an expected value of 3.5 even though no face shows 3.5. Treat E(X) as the long-run per-trial average, and use variance alongside it to judge how far individual results scatter.