Find the expected value, variance and standard deviation of a gamble or decision from its outcomes and probabilities.
How to use this tool
Each possible result with its value — money won or lost, points scored, or any number you are averaging.
As a decimal between 0 and 1 (50% is 0.5). Leave unused outcome rows blank.
Probabilities must add to 1: they have to cover every possible outcome. The calculator rejects anything else.
EV is the long-run average per play. Variance and standard deviation show how wildly individual results swing around it.
Formula & worked example
A gamble paying $100 with probability 0.5 and losing $50 with probability 0.5: EV = 0.5 × 100 + 0.5 × (−50) = 25. Variance = 0.5 × 75² + 0.5 × (−75)² = 5,625, so the standard deviation is 75.
Limitations — read this before you rely on it
Expected value describes the average over many repetitions, not what happens on any single play — a positive-EV gamble can still lose money in the short run, and variance is what measures that risk. Probabilities are only as good as your estimates; garbage probabilities give a precise-looking but meaningless EV.
Frequently asked questions
The probability-weighted average outcome: what you would average per play if you repeated the gamble many times. A positive EV means the gamble favours you in the long run.
Because the outcomes must cover everything that can happen. If they add to 0.8, some 20% outcome is missing and the average is wrong.
Risk. Two gambles can share an EV of $25 while one never strays far from it and the other swings by hundreds — variance (and its square root, the standard deviation) separates them.
Yes, and most casino games are exactly that: a negative EV means the house wins on average, however lucky any one player gets.
Convert to decimals first: 25% is 0.25, 50% is 0.5. The calculator checks that the decimals add to 1.