Expected Value Calculator

What the expected value calculator does

The expected value of a situation with several possible outcomes is the average result you would get in the long run. This calculator works it out from a list of outcomes and the probability of each, adding rows as you need them.

It is the number that tells you whether a bet, a game, or a decision pays off on average. Below is how it works and what it really means.

How to use it

1. Enter each outcome as a value, with its probability as a decimal beside it.
2. Add or remove rows so you have one for every possible outcome, up to twenty.
3. Press Calculate for the expected value, or Reset to clear it. The probabilities must add up to 1.

How the expected value is worked out

Each outcome is multiplied by its probability, and those products are added together:

Expected value = the sum of (each outcome × its probability)

This is a weighted average, where each outcome is weighted by how likely it is. A result that is common pulls the expected value toward itself, while a rare one has little effect, no matter how large it is. Because the probabilities cover every outcome and sum to 1, the weighting is complete, which is why the calculator insists they add up.

The long-run average, not a prediction

The expected value is easy to misread, so here is the careful version. It is not what will happen on any single try, and it need not even be a possible outcome. It is the average you would approach if you repeated the situation a great many times.

The roll of a fair die makes this vivid. Its expected value is 3.5, yet you can never roll a 3.5, since the faces are whole numbers. What the 3.5 means is that over thousands of rolls, the average of the results settles near it. So read the expected value as a long-run tendency, not a forecast of the next outcome.

Where expected value is used

Expected value is how you weigh a decision with uncertain results. In gambling, comparing the expected value of a bet against its cost tells you whether it favours you or the house, and for almost every casino game it favours the house. In insurance, the premium is set above the expected payout, which is how the insurer stays in business.

The same logic guides any choice under uncertainty, from business investments to everyday risks. Working out the expected value of each option puts them on a common footing, so you can see which is best on average, even when any single result is a matter of chance.

A worked example

Take a roll of a fair six-sided die, where you win an amount equal to the face that comes up. Each face, 1 through 6, has a probability of one sixth.

The expected value is each face times its probability, added up: (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 3.5. So over many rolls you would win 3.5 on average per roll, even though no single roll ever pays exactly 3.5. That gap between the average and any actual result is the whole idea of expected value.

Entering your values

Enter each outcome with its probability as a decimal, so a 25 percent chance is 0.25. The probabilities across all your outcomes must add up to 1, since together they have to cover everything that can happen. The expected value is returned once the rows are complete.

Questions people ask

What is expected value?

The long-run average outcome of a situation, found by multiplying each outcome by its probability and adding the results. It is a probability-weighted average.

Can the expected value be an impossible outcome?

Yes. The expected value of a die roll is 3.5, which is not a face you can roll. It is the average over many tries, not a single result.

Why must the probabilities add up to 1?

Because they must cover every possible outcome. If they do not sum to 1, some outcome is missing or mismeasured, and the weighted average would be wrong.

What is expected value good for?

Weighing decisions under uncertainty, such as whether a bet favours you, how to price insurance, or which uncertain option is best on average.

References

A quick note on where the methods here come from. Expected value as the mean of a random variable is set out in the NIST/SEMATECH e-Handbook of Statistical Methods, the US government's public statistics reference. OpenStax Introductory Statistics is a free, widely used textbook covering expected value.

1. NIST/SEMATECH e-Handbook of Statistical Methods (expected value and random variables). https://www.itl.nist.gov/div898/handbook/
2. OpenStax, Introductory Statistics (expected value of discrete random variables). https://openstax.org/details/books/introductory-statistics-2e