Expected Value Calculator
Calculate the expected value of a discrete random variable from values and probabilities. The calculator also checks the probability total and finds variance and standard deviation for a fuller homework-style answer.
What expected value means in a probability problem
Expected value is the long-run average outcome of a random process. It does not always have to be a possible outcome. OpenStax explains expected value in its exact lesson on mean or expected value and standard deviation for random variables.
This calculator is built for student probability tables. You enter each possible value and its probability, and the calculator finds E(X), variance, and standard deviation. That makes it useful for homework questions about games, payouts, random variables, insurance examples, and discrete probability distributions.
Expected value formula
The formula says to multiply each outcome by its probability and add the results. If probabilities do not add to 1, the distribution is incomplete or invalid unless you are intentionally working with partial data.
Expected value is not a prediction for one trial
An expected value describes the average over many repetitions, not what must happen next. If a lottery ticket has an expected value of −$0.40, that does not mean every ticket loses exactly 40 cents. It means that over many tickets, the average loss would approach about 40 cents per ticket under the model.
Frequently asked questions
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Expected value is the long-run average result of a random situation. It is found by multiplying each possible value by its probability and adding the products. It is called “expected” because it describes the average you would expect over many repetitions, not because it must occur in a single trial.
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Yes. Expected value is an average, so it can be a decimal even when every possible outcome is a whole number. For example, the expected number of heads in three fair coin flips is 1.5, even though you cannot actually flip exactly 1.5 heads.
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For a complete probability distribution, probabilities should add to 1. If they add to less than 1, an outcome may be missing. If they add to more than 1, there is likely an error. This calculator warns you when the total is not close to 1 because that usually means the expected value will not represent a valid distribution.
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Probability describes how likely each outcome is. Expected value combines all outcomes and probabilities into one long-run average. A probability might say there is a 30% chance of winning $10. Expected value asks what the average result is after considering every possible outcome.
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Yes. Expected value can be negative if the losses outweigh the gains on average. Many games of chance have negative expected value for the player, meaning the average result over many plays is a loss.