Expected value calculator
Enter possible outcomes and their probabilities to find the probability-weighted average.
Enter outcomes and probabilities
Probabilities must total 100%. Negative outcomes are allowed.
| Outcome | Probability (%) | Remove outcome |
|---|---|---|
Expected value
Average outcome implied by these probabilities.
- Variance
- 38,400
- Standard deviation
- 195.96
How this is calculated
Expected value = sum of (outcome × probability). Variance = sum of (probability × (outcome − expected value)²). Standard deviation is the square root of variance.
300 × 40% + -100 × 60% = 60
Each outcome’s contribution: 120, -60.
Enter a complete, mutually exclusive set of outcomes, with probabilities totaling 100%. Expected value and standard deviation use the same units as your outcomes; variance uses squared units. A probability-weighted average is not a guaranteed result on any single trial. The example represents a 40% chance of +300 and a 60% chance of −100.
Reference: MIT OpenCourseWare · Expected value of discrete random variablesHow to use this calculator
- Enter each possible outcome in the same units. Use negative values for losses and positive values for gains.
- Enter the probability of each outcome as a percentage. Add or remove rows as needed and make the probabilities total 100%.
- Read expected value, variance and standard deviation. Expand the calculation to see each outcome’s contribution.
Example: a gain or a loss
Use Reset example to reproduce these inputs.
- Winning outcome / probability
- +300 / 40%
- Losing outcome / probability
- −100 / 60%
Expected value: 60
The probability-weighted average is 300 × 0.4 + (−100) × 0.6 = 60. Variance is 38,400 and standard deviation is about 195.96. The value 60 summarizes the distribution; a single trial in this example produces either +300 or −100.
Frequently asked questions
Can expected value be negative?
Yes. A negative expected value means the entered distribution has a negative probability-weighted average. Enter losses as negative outcomes. For monetary outcomes, account for costs in the amounts you enter and use one currency consistently.
Why must probabilities add up to 100%?
The rows describe a complete set of mutually exclusive outcomes for one distribution. Their probabilities must sum to 100%. Enter 40 for a 40% probability, not 0.4. If the total is incomplete or exceeds 100%, the calculator clears the result until you correct the probabilities.
Is expected value the outcome of the next trial?
Expected value is an average weighted by the probabilities you entered. It need not be an outcome that can occur. In the example, the mean is 60 while the possible individual results are +300 and −100. Real observations can also differ if the assumed probabilities do not describe the process.
What do variance and standard deviation tell me?
They describe how widely outcomes are spread around the expected value. Variance uses squared outcome units; standard deviation uses the same units as the outcomes. Two distributions can have the same expected value and very different spreads.
Want to understand the calculation?
Practise probability in Python