Expected Value Calculator
Calculate expected value E[X], variance, and standard deviation from outcomes and probabilities. Perfect for probability theory, gambling analysis, investment decisions, and risk assessment with detailed outcome breakdown tables.
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Complete Guide: Expected Value Calculator
Everything you need to know about using this tool effectively
This expected value calculator computes E[X] = sum(x * P(x)) for a discrete random variable. Enter each outcome and its probability, and the tool returns the expected value, variance, and standard deviation. A breakdown table shows each outcome's contribution to the overall result so you can see which events carry the most weight.
Expected value is the probability-weighted average of all possible outcomes and represents the long-run mean if the experiment were repeated many times. Variance measures the spread of outcomes around the expected value, computed as E[(X - E[X])^2]. Standard deviation is the square root of variance and is expressed in the same units as the outcomes, making it easier to interpret than variance. The calculator validates that probabilities sum to one and warns you if they do not, which prevents invalid results from incomplete models.
Statistics Homework
Students solve textbook problems by entering outcome-probability tables and checking E[X], variance, and standard deviation against answer keys.
Gambling Analysis
Players calculate the expected return of bets, lottery tickets, or casino games to quantify the house edge.
Investment Evaluation
Analysts assign probabilities to market scenarios and compute the expected return of an asset or portfolio.
Insurance Pricing
Actuaries determine expected payouts from claim distributions to set fair premium levels.
Business Decision Making
Managers compare expected values across product launch scenarios to choose the highest-value option.
List Your Outcomes
Enter the numerical value for each possible outcome in the left column. Use negative numbers for losses.
Assign Probabilities
Enter the probability of each outcome in the right column. Probabilities must be between 0 and 1 and ideally sum to 1.
Add or Remove Rows
Click add to include more outcomes or delete to remove rows until your model matches the situation.
Click Calculate
The tool computes E[X], variance, and standard deviation and displays a contribution breakdown table.
Copy Results
Copy the output for homework, reports, or presentation slides.
Probabilities must sum to exactly 1 for valid results. The calculator warns you if the total deviates from 1.
Use negative values for costs or losses and positive values for gains so the expected value reflects net benefit.
A high standard deviation means outcomes are spread far from the mean, indicating greater risk or uncertainty.
Pre-built scenarios are helpful for learning. Start with a coin flip or dice roll to understand how the formula works before modeling your own problem.
What does expected value mean in plain language?
Expected value is the average outcome you would get if you repeated an experiment infinitely many times. It is not necessarily a result you will see in any single trial, but rather the long-run center of the distribution.
Why must probabilities sum to 1?
Probabilities represent the complete set of possible outcomes. If they do not sum to 1, some outcomes are missing or your probabilities are incorrect. A total less than 1 means the model is incomplete. A total greater than 1 is mathematically invalid.
What is the relationship between variance and standard deviation?
Standard deviation is the square root of variance. Variance is measured in squared units, which makes it harder to interpret. Standard deviation converts back to the original units so you can compare spread directly to the outcomes.
Can I model losses with this calculator?
Yes. Enter negative values for outcomes that represent losses or costs. The expected value will reflect the net result across gains and losses.
How many outcomes can I enter?
There is no fixed limit. Add as many rows as needed to model your probability distribution. Most practical problems use between 2 and 20 outcomes.
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