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Expected Value Calculator
The expected value is the probability-weighted average outcome — the result you can expect “on average.” Multiply each value by its probability and add them up (Expected Value = Σ (value × probability)). For a die, that is (1+2+3+4+5+6) ÷ 6 = 3.5.
Choose Probability (%) when the odds are known, Frequency for observed results, or Fraction when outcomes are equally likely.
Enter an outcome value (e.g., prize amount or score) on the left and its probability (%) on the right. The tool calculates the expected value and other statistics.
| Outcome value | Probability (%) | |
|---|---|---|
Use probabilities from 0% to 100% that total exactly 100%.
Rows with empty values are ignored
Expected value is the probability-weighted average of outcomes; a single result may differ. Sum and average treat each entered row equally and do not use its probability.
Enter an outcome value (e.g., prize amount or score) on the left and how many times it occurred on the right. The tool weights outcomes by frequency.
| Outcome value | Frequency | |
|---|---|---|
Rows with empty values are ignored
Expected value and average are weighted by frequency. Total is the sum of outcome value × frequency; a single result may differ from the expected value.
Enter an outcome value (e.g., prize amount or score) on the left and its probability as a fraction (e.g., 1/6) on the right.
| Outcome value | Probability (a/b) | |
|---|---|---|
| / | ||
| / | ||
| / |
Use non-negative numerators and positive denominators, with fractions totaling exactly 1.
Rows with empty values are ignored
Expected value is the probability-weighted average of outcomes; a single result may differ. Sum and average treat each entered row equally and do not use its probability.
How to Use
Probability (%) Tab
Enter a value and its probability (%) as a pair. The expected value is calculated once probabilities sum to exactly 100. Use “Fill Remaining” to auto-fill the last empty probability field.
Frequency Tab
Enter a value and the number of times it occurs. The tool uses the total of all frequencies as the denominator to compute each probability, then calculates the expected value. Useful for tallying dice rolls or survey data.
Fraction Tab
Enter a value and its probability as a fraction (e.g. 1/6). The expected value is calculated once all fractions sum to exactly 1. Use “Fill Remaining” to auto-calculate the remaining probability.
Metric Definitions
- Expected ValueThe probability-weighted average outcome. Calculated as the sum of each value multiplied by its probability.
- SumThe total of all entered values (probability not considered).
- AverageThe simple arithmetic mean of the entered values (probability not considered).
- VarianceA measure of how spread out the values are around the expected value. Higher variance means greater spread.
- Std DevThe square root of variance. Expresses spread in the same units as the expected value.
FAQ
How do you calculate expected value?
Multiply each outcome’s value by its probability, then add them all up (Expected Value = Σ (value × probability)). This tool does it automatically once you enter the values with their probabilities, frequencies, or fractions.
What’s the difference between expected value and average?
A simple average treats every value equally, while the expected value weights each value by how likely it is to occur. When all outcomes are equally likely, the two are the same.
Why is a die’s expected value 3.5?
Each face from 1 to 6 appears with probability 1/6, so (1+2+3+4+5+6) ÷ 6 = 3.5. No face shows 3.5, but it is the value you can expect on average.
What are variance and standard deviation?
Variance measures how far the outcomes spread from the expected value, and standard deviation is its square root, expressed in the same units. Larger values mean less predictable results.