Expected Value Calculator
Calculate the expected value for a discrete probability distribution. Enter values and their probabilities.
How to use the expected value calculator
- Enter the possible outcome values, separated by commas or spaces.
- Enter the probability of each outcome, as a decimal (e.g. 0.25 for 25%), in the same order.
- The expected value updates automatically.
What is expected value?
Expected value (also called expectation or mean outcome) is the long-run average result you can expect if you repeat a random experiment many times. It's calculated as the sum of each possible value multiplied by its probability: E(X) = Σ value × probability. For a standard die roll (1-6, equally likely), the expected value is 3.5 — even though you can never actually roll exactly 3.5 on a single roll.
Worked example: an ordinary die has the values 1 to 6, each with probability 1 ÷ 6 ≈ 0.1667. The expected value is (1 + 2 + 3 + 4 + 5 + 6) × 0.1667 = 3.50. Note that 3.5 is not a possible outcome — you can never roll a 3.5. The expected value is the average you converge on over many rolls, not a prediction about the next one.
Frequently asked questions
Why is the expected value of a die 3.5, when you can never roll 3.5?
Expected value is a long-run average, not a possible single outcome. If you roll a die many times and average all the rolls, that average will approach 3.5.
What is expected value used for in practice?
Among other things, in insurance, gambling and investing — to assess whether a bet or an investment is on average profitable over time, given the probabilities of different outcomes.
What happens if the probabilities don't sum to 1?
Then the numbers don't represent a valid, complete probability distribution — either an outcome is missing, or the probabilities are stated incorrectly. The calculator still computes a result, but it should be interpreted with caution in that case.