Expected Value Calculator
Enter a list of outcomes and a matching list of probabilities (or weights, or percents) to find the expected value E(X) = Σ x·p, plus the variance and standard deviation of the distribution.
Example: with Outcomes (x), comma or space separated 1, 2, 3, 4, 5, 6 · Probabilities or weights (same order) 1, 1, 1, 1, 1, 1 → Expected value E(X): 3.5.
- Variance2.917
- Standard deviation1.708
- Probability checkWeights summed to 6 — normalized to 1
Computed by the calculator below using its default values. Change any input to see your own numbers.
E(X) = Σ x·p, the probability-weighted average outcome, not a value any single trial must land on.
Expected value is a weighted average
Expected value answers a plain question: what happens on average if I repeat this many times? You multiply each possible outcome by its probability and add the pieces up, E(X) = Σ x·p. Likely outcomes pull the average toward themselves; rare outcomes barely move it. The result need not be a value you can actually get, since a fair die averages 3.5 even though no face shows 3.5.
Why variance rides along
Two bets can share the same expected value and feel nothing alike. Variance, Σ p·(x - μ) squared, measures how far outcomes spread from the mean, and its square root, the standard deviation, puts that spread back in the original units. A near-zero EV with huge variance is a coin-flip fortune; the same EV with tiny variance is a steady trickle. Reading them together is the whole point.
How it’s calculated
Expected value is E(X) = Σ x_i · p_i over all outcomes. Probabilities are normalized by their sum, so you may enter true probabilities (0.2, 0.3, 0.5) or relative weights (2, 3, 5) and get the same answer. Variance is Σ p_i (x_i - μ) squared using the normalized probabilities; standard deviation is its square root.
Outcomes are treated as the discrete distribution you supply. The tool does not check that your outcomes are exhaustive or mutually exclusive; that modeling is on you.
Expected value of one fair die roll
| Face x | Probability | x times p |
|---|---|---|
| 1 | 1/6 | 0.1667 |
| 2 | 1/6 | 0.3333 |
| 3 | 1/6 | 0.5000 |
| 4 | 1/6 | 0.6667 |
| 5 | 1/6 | 0.8333 |
| 6 | 1/6 | 1.0000 |
The x times p column sums to E(X) = 3.5, the long-run average roll of a fair six-sided die.
Common mistakes
- Probabilities that were meant to sum to 1 but do not. This tool normalizes for you, but an unintended sum changes the weighting from what you expected.
- Mismatched list lengths. Every outcome needs exactly one probability, in the same order.
- Reading EV as a guaranteed result. It is a long-run average; a single trial can land anywhere in the distribution.
Frequently asked questions
How do you find expected value?
Multiply each outcome by its probability and add them: E(X) = Σ x·p. For a fair die that is (1+2+3+4+5+6)/6 = 3.5. Enter your outcomes and their probabilities and the calculator does the weighting.
Can probabilities be entered as weights or percents?
Yes. The tool divides by the total, so 2, 3, 5 gives the same result as 0.2, 0.3, 0.5, and 20, 30, 50 works too. It reports whether your entries already summed to 1.
What is the difference between expected value and average?
For equally likely outcomes they are identical. Expected value generalizes the average by weighting each outcome by its probability, so unlikely outcomes count for less.
Why include variance and standard deviation?
Expected value alone hides risk. Variance and its square root, the standard deviation, show how widely outcomes spread around the mean, which separates a safe bet from a wild one with the same EV.
Can expected value be negative?
Yes. If the probability-weighted losses outweigh the gains, E(X) is negative, the mathematical signature of a bet that loses money on average, like most casino games.