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Expected Value Calculator

Find the average result you'd get over many tries, by weighting each outcome by its chance. Also shows the variance and standard deviation.

Level: Grades 11–12

Printable worksheet

Expected Value Calculator

Inputs

One value per outcome, separated by commas, spaces or new lines. Negative values (losses) are fine.

Decimals from 0 to 1 that add up to exactly 1.

Result

Expected value, E[X]

2

Weighted by their probabilities, the 4 outcomes average out to 2 per trial in the long run.

Variance
291
Standard deviation
17.0587
Number of outcomes
4

See the step-by-step solution ↓

Step-by-step solution

  1. Check that the 4 probabilities add up to 1

    0.4+0.3+0.2+0.1=1

  2. Multiply each outcome by its probability and add the products to get the expected value

    E[X]=−10×0.4+0×0.3+5×0.2+50×0.1=2

  3. For the variance, square how far each outcome is from the expected value. Then weight each square by its probability and add them

    Var(X)=0.4×(−10−2)2+0.3×(0−2)2+0.2×(5−2)2+0.1×(50−2)2=291

  4. Take the square root of the variance. This standard deviation puts the spread back in the outcomes' units

    σ=291=17.0587221092

Test yourself

20 quick questions built from this calculator's formula: work out results, predict what happens when an input changes, and work backward from an answer. Every answer comes with the working.

What this calculator does

Find out what an uncertain choice is worth on average, if you could repeat it many times. That average is called the expected value. It weights each possible outcome by how likely it is. So a large payoff that rarely happens counts for less than a modest one that happens often. Expected value lets you compare uncertain things with one number. Games of chance, insurance prices and business decisions are all compared this way.

Enter each possible outcome and its probability (its chance, from 0 to 1). The calculator returns the expected value, written E[X]. It also gives the variance and standard deviation. These measure how far single results usually stray from the average. Two choices can have the same expected value but very different risk. The standard deviation shows that difference.

Where it's used

Why this math matters outside the classroom:

  • Raffle tickets. A $5 raffle ticket with a 1 in 500 chance at a $1,000 prize has an expected value of $2, a $3 average loss.
  • Insurance pricing. An insurer that sees a 1% chance of a $20,000 claim expects to pay $200 a year on that policy.
  • Game shows. A contestant offered $10,000 or a 50% shot at $25,000 compares the sure amount with an expected $12,500.
  • Business projects. A project with a 60% chance of a $50,000 profit and a 40% chance of a $20,000 loss has an expected value of $22,000.
  • Dice games. A fair six-sided die has an expected roll of (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 3.5.

Grade level and standards: Grades 11–12; Common Core State Standards for Mathematics HSS.MD.A.2, HSS.MD.B.5. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.

How to use it

  1. List the outcome values in the first box, separated by commas, spaces or new lines. Enter losses as negative numbers.
  2. List the probability of each outcome in the second box, in the same order. They must add up to 1.
  3. Read the expected value, the variance and the standard deviation. The working shows each weighted sum.

What the inputs mean

Outcome values (x)
The value of each possible result, such as a prize, a profit or a score. List each outcome only once.
Probabilities (P), in the same order
The chance of each outcome as a decimal from 0 to 1, in the same order as the outcomes. You need one probability per outcome. Together they must add up to 1 (within 0.000000001).

Formula and method

E[X] = Σ xᵢ × pᵢ
Var(X) = Σ pᵢ × (xᵢ − E[X])²
SD = √Var(X)

Here xᵢ is each outcome and pᵢ is its probability. Σ (the Greek letter sigma) means "add them all up". Var(X) is the variance and SD is the standard deviation. The expected value is a weighted average. Multiply each outcome by its probability, then add up the results. The probabilities already add to 1, so you don't need to divide by anything.

For the variance, find how far each outcome is from the expected value, and square it. Then take the weighted average of those squares, using the probabilities. The standard deviation is the square root of the variance. That puts the spread back in the same units as the outcomes. These formulas are for chances you already know. They aren't estimates worked out from a sample of data.

Worked example

A game costs nothing to enter. You lose 10 with probability 0.4, break even (0) with probability 0.3, win 5 with probability 0.2, and win 50 with probability 0.1.

  1. Check the probabilities: 0.4 + 0.3 + 0.2 + 0.1 = 1.
  2. Expected value: (−10 × 0.4) + (0 × 0.3) + (5 × 0.2) + (50 × 0.1) = −4 + 0 + 1 + 5 = 2.
  3. Variance: 0.4 × (−12)² + 0.3 × (−2)² + 0.2 × 3² + 0.1 × 48² = 57.6 + 1.2 + 1.8 + 230.4 = 291.
  4. Standard deviation: √291 ≈ 17.0587.

Result: On average, the game pays 2 per play. But single plays vary widely around that figure (a standard deviation of about 17.06). That's mostly because of the rare large win.

Understanding the result

A positive expected value means you come out ahead on average over many tries. A negative one means you lose on average. It doesn't predict any single result. In the example, no single play ever pays exactly 2.

The standard deviation is a measure of risk. When two options have similar expected values, the one with the smaller standard deviation gives more predictable results.

Assumptions and limitations

  • The list of outcomes is complete, so the probabilities add up to 1.
  • The probabilities are known or well estimated. Expected value is only as good as the probabilities you put in.
  • All the outcome values use the same unit, such as dollars.
  • Expected value ignores how a person feels about risk. Many people sensibly avoid a risky choice with a positive expected value, if they couldn't afford the loss.
  • It works for separate outcomes listed one by one. When outcomes can be any value in a range (a continuous distribution), you need calculus (integration) or a specialized calculator.
  • Up to 1,000 outcomes can be entered.

Frequently asked questions

Can the expected value be a result that never actually happens?

Yes. The expected value of a fair die roll is 3.5, which no single roll can produce. It's a long-run average, not a likely outcome.

What if my probabilities don't add up to exactly 1?

The calculator needs them to add up to 1, give or take a tiny 0.000000001. So rounding to many decimal places is fine. If they fall short, you're probably missing an outcome. If they go over 1, an outcome is probably counted twice, or a probability is too large.

How is expected value different from an ordinary average?

An ordinary average gives every data point equal weight. Expected value weights each possible outcome by its probability. Say you list each value from a data set with a probability of 1 divided by the number of values. Then the two give the same answer.

How do I include the cost of playing a game?

Subtract the cost from every outcome before you enter it, so each value is a net gain or loss. Or subtract the cost from the expected value afterward. Both give the same answer, because the probabilities add up to 1.

How this calculator is checked

Method: Definition of the expected value and variance of a discrete random variable. The calculation is covered by automated tests with known reference values and edge cases. Last updated 2026-10-18. Found a problem? Tell us.

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