What this calculator does
Expected value answers the question: on average, across many repeats, what result should this situation produce? The expected value formula multiplies each possible outcome by its probability and adds those products together, giving a single number that represents the long-run average, even though any individual outcome will usually be different from it.
Expected value shows up anywhere there is uncertainty with known probabilities: a game of chance, an insurance payout, an investment with several possible returns, or a business decision with a range of outcomes. This calculator also reports the standard deviation of the same distribution, which measures how spread out the actual outcomes tend to be around that average.
The formula
Multiply each outcome by its own probability, then add all of those products together to get the expected value, E(X) = Σxᵢpᵢ. Variance is the expected value of the squared outcomes minus the square of the expected value itself, and standard deviation is the square root of that variance.
| Term | Meaning |
|---|---|
| E(X) | Expected value: the probability-weighted average of all possible outcomes. |
| Outcomes | The list of possible results, entered as numbers separated by commas. |
| Probabilities | The chance of each corresponding outcome, entered in the same order as the outcomes list, which should add up to 1. |
The inputs explained
| Field | What to enter |
|---|---|
| Outcomes (comma separated) | The list of possible outcomes, separated by commas, in the same order as their probabilities below. |
| Probabilities (comma separated) | The probability of each outcome, as a decimal (not a percentage), in the same order as the outcomes above; these should sum to 1. |
When to use it
Evaluating a bet or game of chance
A bet with a possible win and a possible loss, each with a known probability, has an expected value that shows whether the bet favours the player or the house on average, over many repeats.
Comparing decisions with uncertain outcomes
When a decision has several possible results, each with an estimated probability and payoff, expected value gives a single comparable number for weighing one option against another.
Pricing insurance or warranty products
Insurers use expected value, the probability of a claim multiplied by its likely payout, as the starting point for what a policy needs to charge on average to cover expected losses.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does expected value change with the probability of each outcome?
The same two outcomes, at different probabilities of losing versus winning.
| Probability of losing / winning | Expected value E(X) | Standard deviation |
|---|---|---|
| 0.5, 0.5 | 5.000 | 15.000 |
| 0.6, 0.4 | 2.000 | 14.697 |
| 0.7, 0.3 | -1.000 | 13.748 |
| 0.8, 0.2 | -4.000 | 12.000 |
| 0.3, 0.7 | 11.000 | 13.748 |
| 0.9, 0.1 | -7.000 | 9.000 |
Questions
Does expected value predict what will actually happen once?
No. Expected value is a long-run average across many repeats, not a prediction for a single instance. A single trial will usually land on one of the actual listed outcomes, not the expected value itself, which may not even be one of the possible results.
What does a negative expected value mean?
It means that, on average across many repeats, the outcome favours loss over gain. A classic example is most casino games, which are deliberately structured to have a negative expected value for the player and a corresponding positive one for the house.
Why do the probabilities need to add up to 1?
Probabilities represent the complete set of possible outcomes, so they must account for 100% of what could happen. If they do not sum to 1, either an outcome is missing or the probabilities were entered incorrectly, and the resulting expected value will not be meaningful.
How is standard deviation useful alongside expected value?
Two situations can share the same expected value while having very different risk. Standard deviation measures how far outcomes typically stray from that expected value, so a low standard deviation means results cluster close to the average, while a high one means individual outcomes vary widely.
For probability calculations involving combined events, see the probability calculator. For converting between odds formats and probability directly, see the odds calculator.