Expected Value Calculator
List each possible outcome (a payoff, a score, a count) and its probability in percent. Leave unused rows at 0. The probabilities must total 100%.
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Formula
- Expected value:
E[X] = sum of x_i x p_i - Variance:
Var(X) = sum of p_i x (x_i - E[X])^2 = E[X^2] - E[X]^2 - Standard deviation:
sd = sqrt(Var(X)) - Probabilities are entered in percent and must total 100% (within 0.01).
Worked example
Outcomes 100 at 10%, 20 at 30%, -5 at 60%. E[X] = 10 + 6 - 3 = 13. E[X^2] = 1000 + 120 + 15 = 1135. Var = 1135 - 13^2 = 966, so sd = 31.0805.
Frequently asked questions
How do you calculate expected value?
Multiply each outcome by its probability and add the results: E[X] = sum of x times p. For 100 at 10%, 20 at 30% and -5 at 60%: 10 + 6 - 3 = 13.
How are variance and standard deviation found?
Variance is the probability-weighted average of squared distance from the mean, equal to E[X squared] minus E[X] squared. Here E[X squared] = 1135, so the variance is 1135 - 169 = 966 and the standard deviation is the square root, 31.08.
Does a positive expected value guarantee a gain?
No. It is the long-run average over many repeats. A single draw gives one of the listed outcomes, and a large standard deviation, like 31.08 against a mean of 13, means individual results vary a lot.
Why must the probabilities add up to 100%?
The outcomes must cover everything that can happen. A total within 0.01 points of 100 is accepted so rounded entries such as 33.33, 33.33 and 33.34 still work. Add a catch-all outcome, for example a value of 0, to cover the rest.