Expected Value Calculator
Calculate expected value, variance and standard deviation from a list of outcomes and probabilities, with a check that the probabilities sum to one and the long-run behaviour explained.
Last reviewed by the Radiatus Cloud team
Need this done properly for your business?
Radiatus delivers secure cloud, DevOps & compliance engineering.
Expected value is an average, not a prediction
The expected value of a fair die is 3.5, which the die can never show. It is the long run average over many repetitions, and it says nothing about any single trial. Treating it as a forecast is the most common misuse: a bet with a positive expected value can still be overwhelmingly likely to lose, and one with a negative expected value can be likely to win. What separates those cases is the variance, which the expected value alone does not capture.
Variance decides whether you survive to reach the average
Two propositions can share an expected value of plus ten and behave nothing alike: one paying ten every time, the other paying a thousand one time in a hundred and losing exactly ten the rest of the time. The law of large numbers guarantees convergence to the mean given enough trials, and the number of trials required grows with the variance. If a run of bad outcomes exhausts your capital before then, the long run average is irrelevant to you, which is why the standard deviation and the worst case both matter as much as the mean.
Where expected value breaks down entirely
For decisions that cannot be repeated, or where a loss is unrecoverable, maximising expected value is the wrong rule. This is the insight behind the St Petersburg paradox, where a game with infinite expected value is worth very little to any real person, and behind the fact that buying insurance has a negative expected value and is still rational. Expected value is the right criterion for repeated, survivable decisions and the wrong one for single, catastrophic ones.
Related tools
- Percentage Calculator — Calculate percentages, percentage change, increase and decrease, and reverse percentages, with the working shown for each result.
- Statistical Calculator — Calculate mean, median, mode, standard deviation, and more.
- Probability Calculator — Calculate probabilities for single and combined events, including conditional probability and Bayes' theorem.
- Scientific Calculator — Perform advanced mathematical operations including trig, log, and exponents.
Frequently Asked Questions
Is a positive expected value always a good bet?
No. A positive expected value with enormous variance can still be overwhelmingly likely to lose over any realistic number of trials, and a run of bad outcomes can exhaust your capital before the average has any chance to assert itself.
Why does variance matter as much as the mean?
Because it determines how many trials are needed before the average dominates. The law of large numbers guarantees convergence eventually, and the number of repetitions required grows with the variance. If you cannot survive the interim, the long run average does not apply to you.
When is expected value the wrong criterion?
For decisions that cannot be repeated and for losses you cannot recover from. Buying insurance has a negative expected value and is entirely rational, because the outcome it protects against is not one you get to average over.
Do the probabilities have to sum to one?
Yes, for the result to be a genuine expected value. The tool reports the sum and, when it is close but not exact, shows what the answer would be if the values were rescaled, so a rounding difference can be told apart from a missing outcome.
What is the St Petersburg paradox?
A game whose expected value is infinite and which nobody would pay much to play. It shows that expected value fails as a decision rule when outcomes are unbounded and the game cannot be repeated, and it prompted the whole idea of utility being different from money.
Privacy & Security
Everything runs in your browser; nothing is uploaded.
How to Use
Enter each outcome with its probability to get the expected value and the spread around it.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
Related Tools
Percentage Calculator
MathCalculate percentages, percentage change, increase and decrease, and reverse percentages, with the working shown for each result.
Statistical Calculator
MathCalculate mean, median, mode, standard deviation, and more.
Probability Calculator
MathCalculate probabilities for single and combined events, including conditional probability and Bayes' theorem.