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Expected Value Calculator
The expected value is the long-run average of a random outcome. Enter each possible outcome and its probability to get the expected value, along with the variance and standard deviation that show how much individual results swing around it.
- Variance
- 2,725
- Standard deviation
- 52.2015
- Sum of probabilities
- 1
Saved setups
Save a set of inputs you reuse — your usual rate, your loan, your room sizes — and load it back in one tap.
Your recent calculations
Results you calculate here are kept on this device so you can come back to them.
Formula
How to use it
- Type the value of each outcome, using negative numbers for losses.
- Type the probability of each outcome in the same order — decimals, fractions such as 1/6, or percentages.
- Check that the probabilities total 1 (or 100%).
- Read the expected value.
Worked examples
Win 100 with probability 0.2, nothing with 0.5, lose 50 with 0.3
- Expected value
- 5
- Variance
- 2,725
- Standard deviation
- 52.2015
- Sum of probabilities
- 1
One roll of a fair six-sided die
- Expected value
- 3.5
- Variance
- 2.9167
- Standard deviation
- 1.7078
A $2 raffle ticket with a 1% chance of a $100 prize
- Expected value
- -1
- Variance
- 99
What it does and does not tell you
Expected value is an average over many repetitions, not a forecast of one. A die has an expected value of 3.5, a number it can never show.
Two bets with the same expected value can carry very different risk; the standard deviation shows the difference. If the probabilities you enter do not total 1 or 100%, no result is given and the sum is shown so you can find the mistake.
Example: a casino bet
A $1 bet on a single number in American roulette pays $35 with probability 1/38 and loses the $1 with probability 37/38. The expected value is 35/38 − 37/38 = −$0.0526, a loss of about 5.3 cents per dollar bet on average.
Questions people ask
What is the expected value of rolling a die?
3.5: each of the numbers 1 to 6 has probability 1/6, and (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 3.5.
What is the expected value of winning 100 with probability 0.2, nothing with 0.5 and losing 50 with 0.3?
5. It is 100 × 0.2 + 0 × 0.5 − 50 × 0.3 = 20 − 15.
What does a negative expected value mean?
On average you lose. A $2 raffle ticket with a 1% chance of a $100 prize has an expected value of −$1: you would lose about a dollar per ticket over many draws.