Expected Value
Calculate the expected value of a discrete random variable online. Enter values and probabilities to compute EV with step-by-step breakdown and charts.
About This Calculator
The Expected Value Calculator computes the expected value (EV) of a discrete random variable. Enter your values and their corresponding probabilities, and the calculator instantly computes E(X) = Σ xᵢ × P(xᵢ). The expected value represents the long-run average outcome when an experiment is repeated many times — a fundamental concept in probability theory, statistics, decision theory, and finance.
The expected value formula is E(X) = Σ xᵢ × P(xᵢ), where xᵢ are the possible values of the random variable and P(xᵢ) are their probabilities. This weighted average of outcomes accounts for the likelihood of each result. For example, a fair six-sided die has E(X) = 1×⅙ + 2×⅙ + 3×⅙ + 4×⅙ + 5×⅙ + 6×⅙ = 3.5.
Regional Notes
India (IN): Expected value is widely used in Indian actuarial science, insurance pricing by IRDAI-regulated companies, and stock market analysis on NSE/BSE. EV calculations help evaluate lottery schemes, insurance policies, and investment opportunities regulated by SEBI.
United States (US): In the US, expected value is fundamental in financial markets (SEC-regulated), casino gaming regulations, insurance underwriting, and business decision analysis. EV modeling is central to portfolio theory and risk management on Wall Street.
United Kingdom (UK): UK regulators including the FCA and Gambling Commission use expected value principles. EV is applied in pension fund risk assessment, actuarial valuations under Solvency II, and cost-benefit analysis for public policy decisions by HM Treasury.
Our calculator supports any number of value-probability pairs. Ensure probabilities are in decimal form (0 to 1) and sum to 1 for a valid probability distribution. The contribution chart visualizes how each value weighted by its probability contributes to the final expected value.
Frequently Asked Questions
What is expected value in statistics?
Expected value (EV) is the long-run average outcome of a random variable, calculated as the sum of each possible value multiplied by its probability of occurrence. It represents what you expect to happen on average when an experiment is repeated many times.
How do you calculate expected value?
To calculate expected value, multiply each possible outcome by its probability, then sum all the products. The formula is E(X) = Σ xᵢ × P(xᵢ). For example, a fair six-sided die has an expected value of 3.5 because each face (1 through 6) has a 1/6 probability: 1×1/6 + 2×1/6 + 3×1/6 + 4×1/6 + 5×1/6 + 6×1/6 = 3.5.
Can expected value be negative?
Yes, expected value can be negative when losses outweigh gains. A negative expected value means the average outcome is a loss, which is why gamblers always lose in the long run against house-favored games. For investments, a positive EV suggests profit potential while negative EV indicates expected loss.
What is the difference between expected value and mean?
The expected value is the theoretical long-run average of a random variable based on known probabilities, while the mean is the arithmetic average of observed data. Expected value is a population concept used in probability theory, whereas the mean describes a sample of actual observations.
How is expected value used in decision making?
Expected value helps quantify the average outcome of decisions under uncertainty. In finance, it is used to evaluate investments by comparing risk-adjusted returns. In insurance, companies use EV to set premiums. In business, EV analysis helps choose between strategies by comparing expected payoffs. A decision with a higher expected value is typically preferred when risk tolerance is neutral.
What does an expected value of 0 mean?
An expected value of zero means the sum of gains and losses balanced by their probabilities equals zero. This is called a fair game. For example, a fair coin flip where you win £1 on heads and lose £1 on tails has an expected value of 0, meaning no profit or loss on average over many trials.
Are probabilities required to sum to 1 for expected value?
Yes, for a valid probability distribution the sum of all probabilities must equal exactly 1. If probabilities do not sum to 1, the expected value calculation may still produce a result, but it will not represent a true expected value. Our calculator displays the probability sum so you can verify your inputs.