Expected Value Calculator
Calculate the expected value of a discrete random variable.
Expected Value Formula
E(X) = ΣxᵢP(xᵢ). Multiply each possible outcome by its probability, then add the products.
Understanding Expected Value
Expected value is the long-run average outcome of a random process when each possible result is weighted by its probability.
Expected Value Example
If a game pays $10 with probability 0.2 and $0 otherwise, its expected payout is $2 before considering any entry cost.
How to Use the Result
Expected value is useful for comparing uncertain outcomes, but it does not guarantee what will happen in a single trial.
Frequently Asked Questions
Can expected value be negative?
Yes, when losses outweigh gains on average.
Does expected value have to be a possible outcome?
No.
Is expected value the same as probability?
No. It combines outcomes with their probabilities.
Interpreting Expected Value
Expected value is the probability-weighted average of all possible outcomes. For a discrete set of outcomes, multiply each possible value by its probability and add the products. The result represents the long-run average you would expect if the same situation were repeated many times.
Expected value is useful in probability exercises, forecasting, games of chance, insurance-style scenarios and decision analysis. For example, an outcome worth 10 with probability 0.6 and an outcome worth 0 with probability 0.4 has an expected value of 6. Probabilities should cover the complete set of outcomes and normally sum to 1.
The expected value is not necessarily an outcome that will occur in any single trial. It also does not describe variability or risk by itself, so two situations can have the same expected value while having very different ranges of possible results.
Methodology and Practical Limits
Expected value is the probability-weighted average of possible outcomes, ΣxP(x) for a discrete variable. It describes a long-run average under the model and need not be an outcome that can occur in a single trial.