Probability · Random variables
Expected Value Calculator
The expected value of a gamble, a game or any uncertain payoff — the probability-weighted average of its outcomes. Enter each outcome and its chance to get E(X), the variance and the standard deviation, with each outcome's contribution laid out and the “balance point” marked on the distribution.
Result
In plain English
Expected value is what you'd get on average if you could run the same gamble over and over — each possible outcome multiplied by its chance, then all added up. It is the single number that summarises a whole spread of possibilities. The catch is in the word “average”: the expected value is often a result you can never actually get on one go.
- expected value, E(X)
- The probability-weighted average: Σ (outcome × probability). The long-run mean over many repetitions — the balance point of the distribution.
- not a single-trial prediction
- A fair die has E(X) = 3.5, a number you can never roll. The expected value tells you about the long run, not the next throw.
- variance & standard deviation
- How spread out the outcomes are around the average. Two gambles with the same expected value can carry wildly different risk.
- contribution
- Each outcome's share of the expected value (outcome × probability). Big, rare payoffs and small, common ones can contribute the same amount.
- a valid distribution
- The probabilities must cover every possibility and add to 1. If yours don't, an outcome is missing — or they're really weights, which this calculator will rescale for you.
Frequently asked
What does expected value actually tell me?
The long-run average outcome if you could repeat the situation many times, with each outcome weighted by its probability. It is not what you'll get on any single try — a fair die's expected value is 3.5, a face you can never actually roll. Think of it as the balance point of all the possibilities.
Do the probabilities have to add up to 1?
Yes. They describe a complete set of mutually exclusive outcomes, so a valid distribution sums to 1 (or 100%). If yours don't, either an outcome is missing, or you've entered relative weights — in which case this calculator divides through by their total and tells you it did.
Is the option with the highest expected value always the best choice?
Not always. Expected value ignores risk: two options with the same average can differ enormously in how spread out their outcomes are. For a one-off, high-stakes decision — where you can't average over many repeats and a bad outcome could wipe you out — the variance, and your tolerance for ruin, can matter more than the mean.
What is the difference between expected value and the most likely outcome?
The expected value is the long-run average if you repeated the gamble many times; the most likely outcome is the single result with the highest probability. They are often different — a lottery’s most likely outcome is “win nothing”, while its expected value is some small positive figure spread across the rare jackpots. Never expect any one trial to land on the expected value; it is a property of the long run, not of the next go.