# Expected Value Calculator

> Find the expected value, variance and standard deviation of a discrete random variable from its outcomes and probabilities, percentages or weights.

Category: Statistics & probability · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/expected-value-calculator

The expected value is the probability-weighted average outcome: what you would average per play over many repetitions.

Also called: EV calculator, Discrete distribution mean calculator.

## Inputs
- Outcomes (values) (default 1, 2, 3, 4, 5, 6)
- Probabilities (default 1/6, 1/6, 1/6, 1/6, 1/6, 1/6)
- Probabilities are (options: Decimals or fractions (sum to 1), Percentages (sum to 100), Relative weights (normalised); default decimal)

## Results
- Expected value E[X]
- Variance
- Standard deviation
- Number of outcomes

## Formula

```
E[X] = Σ xᵢpᵢ;  Var(X) = Σ pᵢ(xᵢ − E[X])²;  SD = √Var(X)
```

Where:
- xᵢ: each outcome
- pᵢ: its probability

## Worked example
1. E[X] = (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 21 ÷ 6 = 3.5.
2. Var = (6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25) ÷ 6 = 17.5 ÷ 6 = 2.916667; SD = 1.7078.

**Result:** A fair die has an expected value of 3.5 with a standard deviation of about 1.71.

## What the result means

A negative expected value for a bet means you lose on average in the long run. The standard deviation shows how far single results typically stray from that average.

## Assumptions and limits
- Outcomes are mutually exclusive and the probabilities cover every possibility.
- Expected value ignores risk preferences: two choices with the same EV can differ greatly in how much you could lose.

## Questions

### Does the expected value have to be a possible outcome?

No. A die never shows 3.5; it is a long-run average.

### How do I value a lottery ticket?

List each prize (net of the ticket price) with its probability, including the “lose the stake” outcome; the EV is almost always negative.

## Related calculators
- [Probability Calculator](https://calculators.geekmidst.com/calculators/probability-calculator)
- [Conditional Probability Calculator](https://calculators.geekmidst.com/calculators/conditional-probability-calculator)
- [Hypergeometric Probability Calculator](https://calculators.geekmidst.com/calculators/hypergeometric-probability-calculator)

Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/expected-value-calculator