Last updated: 2026-06-04

What is expected value?

Expected value (E[X]) is the probability-weighted average of all possible outcomes of a random variable. For a discrete distribution with outcomes x₁, x₂, …, xₙ occurring with probabilities p₁, p₂, …, pₙ, the expected value is:

E[X] = Σᵢ pᵢ · xᵢ

Intuitively, expected value answers: "If I ran this experiment a very large number of times, what would the average outcome be?" The law of large numbers guarantees that the sample mean converges to E[X] as the number of independent trials increases.

Formula and worked example

Consider a game with three outcomes:

Expected value: E[X] = (10)(0.50) + (−5)(0.30) + (20)(0.20) = 5.00 − 1.50 + 4.00 = 7.50

The payoff structure is asymmetric, but over many repetitions the average outcome converges to $7.50 per trial. A casino offering this game would lose money in the long run.

Variance measures spread — how far outcomes deviate from the expected value on average, in squared units:

σ² = Σᵢ pᵢ · (xᵢ − E[X])² = 0.50×(10−7.5)² + 0.30×(−5−7.5)² + 0.20×(20−7.5)² = 0.50×6.25 + 0.30×156.25 + 0.20×156.25 = 3.125 + 46.875 + 31.250 = 81.25

Standard deviation (σ = √81.25 ≈ 9.01) restores the original units and represents a typical deviation from the expected value in a single trial.

Why variance matters alongside expected value

Two distributions can share the same expected value while differing dramatically in variance. A bond paying $100 with certainty (E[X] = 100, σ = 0) and a lottery paying$200 with probability 0.5 and $0 otherwise (E[X] = 100, σ = 100) have identical expectations but very different risk profiles. Rational decision-making under uncertainty requires considering both.

Fair game definition

A fair game is one in which E[X] = 0: the expected gain is zero. In a fair game, the house neither profits nor loses over the long run, and the player faces only risk. Most gambling activities are designed so that E[X] < 0 for the player, providing a systematic profit for the operator. Enter the game's outcomes and probabilities to verify whether it is fair (E[X] = 0), favorable (E[X] > 0), or unfavorable (E[X] < 0).

The law of large numbers

The strong law of large numbers states that for independent, identically distributed random variables with finite expected value, the sample mean converges almost surely to E[X] as the number of observations grows. This is the theoretical basis for insurance, financial pricing models, and any long-run planning that relies on averaging across many events.

Critically, the law says nothing about any single observation or small sequence of trials. A game with E[X] = +5 can produce a run of negative results over dozens of plays. The standard deviation quantifies exactly this short-run uncertainty: roughly 68% of individual outcomes fall within one σ of E[X], and roughly 95% within two σ.

Using this calculator

Enter outcomes and their corresponding probabilities as comma-separated values. The counts must match. Probabilities must be non-negative and sum to 1; the calculator warns if they deviate from 1 by more than 0.001. Outcomes can be any real numbers, including negatives.

The probability distribution chart plots each outcome's probability as a bar, giving an immediate visual sense of where the probability mass is concentrated.