Expected Value Calculator
Find the expected value from outcomes and probabilities.
Comma-separated
Should sum to 1
Results update as you type.
How it works
Expected value is the long-run average outcome of a random process, and this calculator computes it from a list of possible outcomes and their probabilities. You enter the outcome values and the matching probabilities, and it multiplies each outcome by its probability, sums the products, and returns the expected value, along with the total probability so you can confirm it equals one. Expected value answers the question: if you repeated this random situation over and over, what would the average result be? It is not necessarily a value that can actually occur on any single trial; rather, it is the probability-weighted centre of the distribution. Expected value is central to decision-making under uncertainty: gamblers and casinos use it to evaluate bets, insurers use it to price policies, investors use it to weigh risky returns, and economists use it in utility theory. A positive expected value on a bet means it is favourable in the long run, while a negative one means it loses on average, which is exactly how casino games are designed. For a valid calculation, the probabilities should be non-negative and sum to one. Enter the outcomes and their probabilities to compute the expected value.
E(X) = Σ(value × probability)
The expected value of a random variable is the sum of each possible outcome multiplied by its probability. It represents the long-run average result you would expect if the random process were repeated many times. The calculator also totals the probabilities so you can check they sum to one, as they should for a complete and valid probability distribution.
Source:Probability theory (expected value)
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Worked example
How to use this calculator
- 1
Enter your values
Enter outcome values and probabilities.
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Expected value updates instantly as you type — no button to press.
Frequently asked questions
What is expected value?
Expected value is the probability-weighted average of all possible outcomes of a random variable. It represents the long-run average you would get by repeating the random process many times. Calculated as the sum of each outcome times its probability, it is a single number that summarises the central tendency of an uncertain situation, even if it is not an outcome that can literally occur.
How do I calculate expected value?
Multiply each possible outcome by its probability, then add all those products together. For a die roll paying the number shown, multiply each face value by one-sixth and sum them to get an expected value of 3.5. Enter your outcomes and their probabilities into the calculator, and it performs this weighting and summation for you automatically.
Should the probabilities add up to one?
Yes. For a complete and valid probability distribution, the probabilities of all possible outcomes must sum to exactly one, since one of them is certain to happen. The calculator reports the total so you can check this. If your probabilities do not sum to one, you may be missing an outcome or have a rounding or entry error.
Can expected value be an impossible outcome?
Yes. The expected value is an average, not a guaranteed result, so it need not equal any actual outcome. The expected value of a fair six-sided die is 3.5, even though you can never roll a 3.5. It represents the balance point of the distribution, the long-run mean, rather than a value you will observe on a single trial.
How is expected value used in decisions?
Expected value guides choices under uncertainty by summarising each option's average payoff. A decision with higher expected value is better on average over many repetitions. Insurers, investors, and gamblers all compare expected values, though real decisions also weigh risk and the variability of outcomes, since a high average can hide the chance of a catastrophic loss.