Expected Value Calculator
Type one outcome per line as a value and a probability, such as 100, 25% or 10, 1/4, and the calculator works out the expected value, the variance, the standard deviation, the chance of a gain or a loss and the most likely outcome. Add a cost to play to see the net result of a bet or a raffle ticket, or switch on weights to enter counts instead of probabilities. Everything runs in your browser.
| Value | Probability | Value x probability |
|---|---|---|
| 1 | 0.166667 | 0.166667 |
| 2 | 0.166667 | 0.333333 |
| 3 | 0.166667 | 0.5 |
| 4 | 0.166667 | 0.666667 |
| 5 | 0.166667 | 0.833333 |
| 6 | 0.166667 | 1 |
The expected value is the sum of each value times its probability. It is the long-run average per play, not what happens on one play. The variance is the average of the squared values minus the square of the expected value, and the standard deviation is its square root. Write a probability as a decimal, a percent or a fraction. With a cost to play, the cost is taken from every value, so a negative expected value means you lose money on average. The presets are examples, and real odds depend on the game. Your outcomes stay in your browser.
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Student ToolsAbout the Expected Value Calculator
The expected value of a random outcome is the sum of each value times its probability. It is the average result per play if you could repeat the game many times, and it is not what happens on any one play. A fair six-sided die has an expected value of 3.5, even though you can never roll a 3.5. The variance is the average of the squared values minus the square of the expected value, and the standard deviation is its square root, which tells you how far results typically stray from the average. A bet with an expected value of zero is fair, a negative value means you lose on average, and a positive value means you gain on average. A 500 prize raffle with 1,000 tickets sold at 2 each has an expected value of 500 / 1000 - 2 = -1.50 per ticket. A single number on a 37 slot roulette wheel paying 35 to 1 has an expected value of about -0.027 per unit staked. Probabilities can be written as decimals, percents or fractions, and they must add up to one unless you choose weights, in which case they are scaled for you. The tool describes the maths of the numbers you enter and does not check whether the odds you typed are realistic.
How do I calculate expected value?▾
Multiply each outcome by its probability and add the results. For a fair six-sided die that is (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5.
What does a negative expected value mean?▾
You lose money on average each time you play. Casino games and lottery tickets usually have a negative expected value for the player, even though a single play can win.
What is the difference between expected value and probability?▾
A probability is the chance of one outcome. The expected value combines every outcome with its size, so it tells you the average result and not the chance of one result.
Do the probabilities have to add up to 1?▾
Yes, for a complete set of outcomes. The calculator tells you if they do not. If you only have counts or weights, switch on the weights option and it scales them to add up to one.
Is my data uploaded?▾
No. The calculation runs in your browser and nothing you enter is sent anywhere.