Expected Value
Weighted average outcome.
What does Expected Value mean?
Expected value (EV) is the long-run average outcome of a random variable when the experiment is repeated many times. It represents the weighted average of all possible values, where each value is weighted by its probability of occurring. EV is a fundamental concept in probability, statistics, decision theory, and finance — it helps you evaluate bets, investments, and any situation involving uncertainty.
How to calculate Expected Value
Expected value is calculated with the formula: EV = sum of (value_i x probability_i) for all outcomes. Each probability should be expressed as a decimal (e.g., 50% = 0.50), and all probabilities must sum to 1. For example, if you can win $100 with 50% chance, win $200 with 30% chance, or lose $50 with 20% chance, the EV = (100 x 0.50) + (200 x 0.30) + (-50 x 0.20) = 50 + 60 - 10 = $100. The standard deviation measures how spread out the outcomes are around the expected value.
FAQ
A positive expected value means that, on average, you will gain over the long run. For example, an EV of $100 means you can expect to earn $100 on average each time the scenario plays out. However, individual outcomes can still vary widely — the standard deviation tells you how much.
Yes, for a valid probability distribution all probabilities must sum to 100% (or 1.0 in decimal form). If they do not, the expected value calculation will still produce a number, but it will not be mathematically meaningful. Always double-check that your probabilities represent all possible outcomes.
Investors use expected value to estimate the average return of an investment by weighting each possible return by its likelihood. For instance, if a stock has a 60% chance of gaining 20% and a 40% chance of losing 10%, the expected return is (0.60 x 20%) + (0.40 x -10%) = 8%. This helps compare investments with different risk profiles.
The mean (average) is calculated from observed data, while expected value is a theoretical prediction based on probabilities. When you have a known probability distribution, you compute expected value. When you have a dataset of actual observations, you compute the mean. As the number of observations grows, the mean tends to converge toward the expected value.
Standard deviation measures the risk or variability of outcomes around the expected value. Two scenarios can have the same EV but very different risk levels. A low standard deviation means outcomes cluster tightly around the EV, while a high standard deviation means outcomes are more spread out and less predictable.
Related calculators
- Standard Deviation— Spread of values.
- Variance— Squared dispersion measure.
- Probability— Likelihood of outcome.
- Mean— Average value.