Expected Value Calculator
The long-run average of a set of outcomes weighted by their probabilities — with the variance, standard deviation and a check that the probabilities sum to one.
Expected value is Σ pᵢ xᵢ: each outcome multiplied by its chance, summed.
How the expected value calculator works
Expected value is Σ pᵢ xᵢ: each outcome multiplied by its chance, summed. It is not the most likely outcome and need not be a possible outcome at all — the expected value of a die roll is 3.5.
The variance Σ pᵢ(xᵢ − E)² measures how far from that average individual results tend to land, which is what separates a fair bet from a survivable one.
Formula: E[X] = Σ pᵢ xᵢ; Var = Σ pᵢ (xᵢ − E)²
Worked examples
| Inputs | Expected value | Note |
|---|---|---|
| 70% lose 1, 25% win 5, 5% win 20 | 1.55 | EV = +1.55 |
| A fair die | 3.5 | EV = 3.5 |
| A coin flip for ±10 | 0 | EV = 0 |
FAQFrequently asked questions
What is expected value?
The probability-weighted average outcome — what the average result converges to over many repetitions.
Is it the most likely outcome?
No. A die's expected value is 3.5, which it can never roll.
What does a negative expected value mean?
That repeating the bet loses money on average. Every commercial gambling product has one.
Why does the variance matter?
Because a positive expectation with enormous variance can still ruin you before the long run arrives. That is the whole subject of risk management.
Do my probabilities have to sum to 100%?
They should. If they do not, the calculator normalises them and says so.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Australian Curriculum (ACARA) — Mathematics — the terms and methods taught in Australian schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.