How do you calculate expected value?

Multiply each outcome by its probability and add the products: E(X) = Σ x·P(x). For a fair die, (1 + 2 + 3 + 4 + 5 + 6) × 1/6 = 3.5.

Can the expected value be a value X never takes?

Yes. A fair die has an expected value of 3.5 even though no roll shows 3.5. It is an average, not a prediction of one outcome.

How is variance of a discrete distribution calculated?

Var(X) = Σ x²·P(x) − [E(X)]². The standard deviation is the square root of the variance.

What if my probabilities do not add to 1?

Check for a missing outcome or rounding. The calculator accepts totals within 0.001 of 1 and shows an error otherwise.

Expected Value Calculator with Steps

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Expected Value Calculator

The expected value E(X) is the long-run average outcome of a random process: the sum of each outcome multiplied by its probability. Enter the possible values of X and their probabilities in the same order, and the calculator builds the x·P(x) table, adds it up and also returns the variance and standard deviation.

Probabilities must add up to 1. The variance uses the shortcut Var(X) = Σx²P(x) − [E(X)]², and the standard deviation is its square root. A game with a negative expected value loses money on average, however good individual plays look; an insurance premium is priced above the expected payout for the same reason.

Typical uses include dice and card games, lottery and raffle tickets, number of goals or defects per unit, and decision problems where each option has known payoffs. For counts that follow a standard model you can skip the table: the binomial and Poisson calculators give the mean and variance directly.

Outcomes x

Separate values with commas, spaces or new lines.

Probabilities P(x)

One probability per outcome, in the same order. They must add up to 1 (fractions such as 1/6 are not accepted; use 0.1667).

Result and step-by-step solution

Expected value E(X)

2.05

Variance = 1.4475, standard deviation = 1.203121.

Multiply each outcome by its probability
x P(x) x · P(x) x² · P(x)
0 0.1 0 0
1 0.25 0.25 0.25
2 0.3 0.6 1.2
3 0.2 0.6 1.8
4 0.15 0.6 2.4
Total 1 2.05 5.65
Expected value
E(X) = Σ x · P(x) = 2.05
Variance
Var(X) = Σ x² · P(x) − [E(X)]² = 5.65 − 2.05² = 1.4475
Standard deviation
σ = √Var(X) = 1.203121

x	P(x)	x · P(x)	x² · P(x)
0	0.1	0	0
1	0.25	0.25	0.25
2	0.3	0.6	1.2
3	0.2	0.6	1.8
4	0.15	0.6	2.4
Total	1	2.05	5.65

Frequently asked questions

How do you calculate expected value?: Multiply each outcome by its probability and add the products: E(X) = Σ x·P(x). For a fair die, (1 + 2 + 3 + 4 + 5 + 6) × 1/6 = 3.5.
Can the expected value be a value X never takes?: Yes. A fair die has an expected value of 3.5 even though no roll shows 3.5. It is an average, not a prediction of one outcome.
How is variance of a discrete distribution calculated?: Var(X) = Σ x²·P(x) − [E(X)]². The standard deviation is the square root of the variance.
What if my probabilities do not add to 1?: Check for a missing outcome or rounding. The calculator accepts totals within 0.001 of 1 and shows an error otherwise.

Last reviewed: October 7, 2026. Calculations run in your browser and were validated against SciPy.

Expected Value Calculator

Result and step-by-step solution

Frequently asked questions

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