How to Calculate Expected Value

Expected value equals the sum of each outcome's probability times its payoff: multiply every payoff by its probability and add the products.

The Expected Value formula

Expected value = Σ (probability of an outcome × payoff of that outcome), where the probabilities must sum to 1

Calculator

Enter three outcomes with their probabilities and payoffs to get the expected value and the net gain.

Probability of outcome 1

Entered as a percent. All three must add to 100.

Payoff of outcome 1

Probability of outcome 2

Payoff of outcome 2

Probability of outcome 3

Payoff of outcome 3

Losses are negative payoffs.

Cost to take the gamble

What you pay up front. Set it to zero if entering is free.

Expected value

$85

Repeated many times this gamble averages $85 per try, which is often not one of the possible outcomes.

Probabilities total: 100%
Probability check: Probabilities sum to 1
Weighted payoffs: $100, $45, −$60
Expected net gain after cost: $15
Verdict: Worth taking

How to calculate Expected Value, step by step

1
List every possible outcome. Write out each payoff, using negative numbers for losses.
2
Attach a probability to each. Convert percentages to decimals and confirm they sum to 1, otherwise an outcome is missing.
3
Multiply each payoff by its probability. This weights every outcome by how likely it is to happen.
4
Add the weighted payoffs. The total is the expected value, the average result if the situation repeated many times.
5
Compare it to the cost. Subtract what you pay to take the gamble; a positive expected net gain means it pays off on average.

Worked example: Expected Value

An investment has a 25% chance of gaining $400, a 45% chance of gaining $100, and a 30% chance of losing $200. The probabilities are complete: 0.25 + 0.45 + 0.30 = 1. Expected value = (0.25 × 400) + (0.45 × 100) + (0.30 × (−200)) = 100 + 45 − 60 = $85. If entering costs $70, the expected net gain is 85 − 70 = $15, so it pays off on average.

Expected Value questions

What if the probabilities do not add to 1?

You have either missed an outcome or made an arithmetic error. Every possible result must be listed, and the probabilities must total exactly 1 before you weight the payoffs.

Is expected value the outcome you should expect?

Usually not. It is a probability-weighted average that often is not even one of the possible results, so it describes the long-run average over many repetitions rather than any single outcome.

Why would someone refuse a bet with a positive expected value?

Risk aversion. Because of diminishing marginal utility of income, the utility lost from a possible loss can outweigh the utility gained from an equal-sized win, so a risk-averse person may decline.

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Last updated August 1, 2026

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