Expected Value Calculator
Find the long-run average outcome of a probabilistic scenario. Enter up to three possible outcomes and their probabilities to compute the expected value — the foundation of decision theory and risk analysis.
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Formula
E(X) = Σ xᵢ × P(xᵢ)
Multiply each possible outcome by its probability (expressed as a decimal), then sum all the products. The probabilities should add up to 100%. The expected value is the theoretical long-run average — it may not equal any individual outcome, but it is the best single-number prediction for the average result over many repetitions.
How to use the Expected Value Calculator
- 1
Enter your outcome 1
- 2
Enter your probability 1 (%)
Value should be in %.
- 3
Enter your outcome 2
- 4
Enter your probability 2 (%)
Value should be in %.
- 5
Enter your outcome 3
- 6
Enter your probability 3 (%)
Value should be in %.
- 7
Read your results instantly
Results update in real time as you type.
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What expected value tells you
Expected value is the probability-weighted average of all possible outcomes. For the defaults {100 at 30%, 50 at 50%, 0 at 20%}, the expected value is 55. This means if you repeated this gamble thousands of times, your average outcome would converge to 55.
Expected value is central to gambling, insurance, finance, and any decision made under uncertainty. A bet is considered fair if its expected value equals its cost. A positive expected value means the bet favors you on average; negative means it favors the house.
Expected value vs. most likely outcome
Expected value and the most probable outcome are often different. In the default example, the most likely single outcome is 50 (50% probability), but the expected value is 55. The higher outcome (100) at 30% pulls the average up significantly.
This is why expected value is more useful for repeated decisions than for one-time choices. For a single high-stakes decision, you might also care about the variance (risk) and the possibility of extreme outcomes — a strategy with a higher expected value but greater downside risk may not always be preferable.
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Applications in business and finance
Businesses use expected value for project valuation: multiply each possible payoff by its probability and sum. Venture capitalists evaluate startup investments this way, knowing most fail but a few succeed spectacularly. Insurance companies set premiums by computing the expected payout plus overhead and profit margin.
In finance, expected return is the expected value of an investment's return distribution. Portfolio theory tries to maximize expected return for a given level of variance (risk), formalizing the trade-off between these two quantities.
Tips & Insights
Probabilities must sum to 100%
If prob1 + prob2 + prob3 ≠ 100%, the expected value will be incorrect. Always verify your probabilities sum to exactly 100% before interpreting the result.
Negative outcomes are valid
Outcomes can be negative (losses, costs, penalties). Enter negative numbers to represent losses. The expected value can also be negative, indicating an average loss.
Expected value ignores risk
Two scenarios with the same expected value can have very different risk profiles. A certain gain of $50 and a coin flip for $0 or $100 both have EV = $50, but most people prefer the certain gain. Expected value alone does not capture this preference.
Worked Examples
Simple lottery ticket
Expected value: $12.00. If the ticket costs more than $12, the expected value is negative — the lottery is a losing proposition on average.
Business project evaluation
Expected value: $175,000. The project has a positive expected value, suggesting it is worth pursuing if the downside risk is acceptable.
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Frequently Asked Questions
What is expected value?
Expected value is the probability-weighted average of all possible outcomes of a random variable. It represents the long-run average result if the situation were repeated many times.
Can expected value be negative?
Yes. A negative expected value means the average outcome is a loss. Most casino games have negative expected value for the player, which is how casinos profit over time.
Why should probabilities add up to 100%?
Probabilities represent a complete set of possibilities — something must happen. If they sum to less than 100%, some outcome is unaccounted for. If they sum to more, the scenario is over-counted.
Is expected value the same as the average?
Expected value is a theoretical average based on probabilities. The sample mean is an observed average from actual data. They converge as the number of observations grows (the law of large numbers).
How is expected value used in insurance?
Insurers calculate the expected payout for each policy (probability of claim × claim amount), then charge premiums that exceed this expected cost plus administrative expenses, generating profit on average across a large pool of policies.
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