Expected Value Calculator
Enter outcomes and their probabilities to evaluate the expected value of a discrete random variable. Add or remove rows as needed.
| Outcome (x) | Probability P(x) | Actions |
|---|---|---|
Expected Value
3.6000
Σ x · P(x)
Probability Sum
1.0000
Σ P(x)
Normalized E[X]
3.6000
Expected value if probabilities sum to 1
How to Use This Calculator
- List each possible outcome and its associated payoff or value.
- Enter the probability of each outcome. Probabilities do not need to sum to 1, but the sum is shown for reference.
- Add or remove rows to match your scenario.
- Review the expected value, total probability, and normalized expectation.
Formula
E[X] = Σ xi · P(xi)
The expected value (mean) represents the long-run average outcome if the experiment were repeated many times. Probabilities should represent a complete distribution when possible; otherwise, the normalized expected value divides by the probability sum.
Full Description
Expected value is a cornerstone of probability, finance, and decision theory. It combines outcomes and their likelihoods to reveal the average payoff you can expect over repeated trials. Positive expected value suggests a profitable bet; negative expected value warns that losses will dominate.
This calculator supports any discrete distribution, from gambling bets to business forecasts. Use it to compare options, evaluate risks, and communicate probabilistic thinking to stakeholders.
Frequently Asked Questions
Do probabilities have to sum to 1?
Ideally yes, but the calculator handles partial totals by displaying the sum and computing a normalized expected value for convenience.
What if I only know frequencies?
Convert frequencies to probabilities by dividing each count by the total number of observations, then enter those probabilities here.
Can I model uncertain probabilities?
Yes. Try different probability scenarios and compare the resulting expected values to understand sensitivity to uncertainty.
How does this relate to variance?
Variance requires squaring deviations from the mean. Extend the table with an additional column for (x − E[X])² · P(x) to calculate it manually.