Expected Value Calculator
Compute the expected value, variance and standard deviation of a discrete outcome from up to four value-and-probability pairs.
How it works
E(X) = Σ xᵢ · pᵢ · Variance = Σ pᵢ · (xᵢ − E(X))² · (probabilities rescaled to sum to 1)
The expected value is the probability-weighted average of every possible outcome — the long-run mean if the situation were repeated many times. Each value is multiplied by its probability and the products are summed. It anchors decisions under uncertainty, from insurance pricing to betting: a positive expected value favours you on average, a negative one works against you. The variance and standard deviation then describe how much any single result is likely to swing around that average.
Worked example
A bet pays +100 with probability 25%, 0 with probability 50%, and −40 with probability 25%. E(X) = 100·0.25 + 0·0.50 + (−40)·0.25 = 25 + 0 − 10 = 15, so the average outcome is +15 per play.
Sources: Expected value — Wikipedia
Frequently asked questions
Do the probabilities have to add up to 100%?
Ideally yes, since they should cover every outcome. If your entries do not sum to 100%, this calculator rescales them proportionally so the expected value still makes sense, and it shows the total you entered.
What does a negative expected value mean?
It means that, averaged over many repetitions, you lose. A casino game with a negative expected value for the player is exactly how the house edge works — individual wins happen, but the long-run average is a loss.
Is the expected value a result I should actually expect?
Not necessarily on any single trial — it may not even be a possible outcome. It is the average over many trials, so pair it with the standard deviation to understand the spread of what can happen.
Related calculators
For education — check the assumptions (sample size, independence, distribution) before using these results in real decisions.