About the Expected Value Calculator
Expected value is the average result you would get if you could repeat a chancy situation again and again. For a game with a 10 percent chance of winning 100 and a 30 percent chance of winning 20, the expected value combines each prize with its probability. This calculator takes a list of outcomes and their probabilities, one per line, and gives the expected value, the variance and the standard deviation. Probabilities can be typed as decimals, percentages or fractions, and negative values are allowed for losses. It also warns you when the probabilities do not add up to 100 percent, which usually points to a missing outcome.
Updated October 2026
How to use the Expected Value Calculator
- 1Enter one outcome per line as a value and a probability, such as 100, 0.1.
- 2Use decimals, percentages like 25%, or fractions like 1/4.
- 3Enter losses as negative values.
- 4Check that the probabilities add up to 100 percent.
- 5Read the expected value, variance and standard deviation.
A worked example
A coin game pays 10 for heads and costs 6 on tails. With a fair coin that is 10 × 0.5 − 6 × 0.5 = 2 per play. The variance is 0.5 × (10 − 2)² + 0.5 × (−6 − 2)² = 64, so the standard deviation is 8. The expected gain is small compared with that spread, so a short run could easily lose money even though the game favours you in the long run.
In many problems the hard part is listing the outcomes. For dice, cards and lotteries, the Permutation and Combination Calculator counts the ways each prize can occur, and dividing by the total number of equally likely outcomes gives each probability. For an insurance policy you might list no claim, a small claim and a large claim with their yearly probabilities. The expected value, multiplied by the number of policies, then estimates the total claims an insurer should prepare to pay.
Entering outcomes
- One line for each outcome, value first and probability second.
- Probabilities: 0.25, 25% or 1/4 all work.
- Use a minus sign for losses or costs.
- Include the outcome of nothing happening, with value 0.
- Check the total probability line.
Limits
Expected value assumes that the probabilities are right and that you can repeat the situation many times. One-off decisions, rare disasters and cases where the amounts are large compared with your wealth call for more careful thinking about risk. Treat the number as a guide, not a guarantee.
Related tools
The Probability Calculator handles single events, the Binomial Distribution Calculator covers repeated trials, the Permutation and Combination Calculator counts arrangements and the Standard Deviation Calculator describes data you have already collected.
Features
- Expected value, variance, standard deviation
- Probabilities as decimals, percentages or fractions
- Warns when probabilities don't sum to 1
- Negative values for losses
- Any number of outcomes
- Private and instant
Frequently asked questions
What is expected value?
The sum of each outcome multiplied by its probability. It is the long-run average per trial. With outcomes 100 (10%), 20 (30%) and 0 (60%), the expected value is 10 + 6 + 0 = 16.
Is expected value what will happen?
No. It is an average over many repeats, and any single trial will usually be different. In the game above you will never win exactly 16, but over thousands of plays you will average close to it.
How do I use it for betting or games?
Enter the winning and losing amounts net of your stake with their probabilities. If the expected value is negative, you lose money on average. A raffle ticket costing 5 with a 1 in 1,000 chance of winning 2,000 has an expected value of 2 − 5 = −3.
What do variance and standard deviation add?
They measure how spread out the outcomes are around the expected value. Two games can have the same expected value but very different risk. A higher standard deviation means a bumpier ride.
What should the probabilities add up to?
1, or 100 percent, since one of the listed outcomes must happen. The tool tells you if they don't, so you can find a missing outcome or a typing error.
Can I use it for insurance or business decisions?
Yes, as a first step. Insurers weigh claim sizes by probability, and businesses compare projects by expected profit. Real decisions also consider risk, because people may prefer a smaller certain result to a larger uncertain one.