Expected Value Calculator
Find the expected value, the long-run average, of a bet, game or decision with several outcomes, plus the variance of the distribution, exactly with fractions.
About the Expected Value Calculator
Enter the possible outcomes and their probabilities (decimals, fractions or percentages that add up to 1 or 100). The calculator gives the expected value, the variance and the standard deviation, exactly when you use fractions.
Expected value weights each outcome by its probability. A raffle ticket costing 10 with a 1-in-100 chance of winning 500 has an expected value of 500 × 0.01 − 10 = −5: on average you lose 5 per ticket.
The formula
E[X] = Σ xᵢ × pᵢ
- Variance
- Σ pᵢ(xᵢ − E[X])²
Worked example
Win 100 with probability 0.1, 20 with 0.3, lose 10 with 0.6
- E = 100 × 0.1 + 20 × 0.3 − 10 × 0.6
- = 10 + 6 − 6 = 10
- Variance 1,080; standard deviation 6√30 ≈ 32.86
Questions people ask
Does a positive expected value guarantee profit?
No, only on average over many tries. Any single try can still lose.
How do I include the cost of playing?
Subtract the cost from each outcome value.
What does the standard deviation tell me?
How spread out the results are: a large one means single tries swing far from the average.