Enter Outcomes and Probabilities
Use probabilities that add to 1. Blank rows are ignored automatically.
Example Data Table
| Scenario | Outcome | Probability | Weighted Contribution |
|---|---|---|---|
| Low return | 1 | 0.10 | 0.10 |
| Moderate return | 3 | 0.20 | 0.60 |
| Average return | 5 | 0.30 | 1.50 |
| Strong return | 7 | 0.25 | 1.75 |
| Peak return | 10 | 0.15 | 1.50 |
| Total | 1.00 | 5.45 | |
In this example, the mathematical expectation equals 5.45.
Formula Used
Mathematical expectation measures the weighted average of all possible outcomes in a discrete distribution.
Expected value formula: E(X) = Σ [x × P(x)]
Variance formula: Var(X) = Σ [P(x) × (x - E(X))²]
Standard deviation formula: σ = √Var(X)
Here, x represents each outcome and P(x) represents its probability. The probabilities must sum to 1 for a valid result.
How to Use This Calculator
- Enter each possible outcome in the outcome fields.
- Enter its matching probability beside that outcome.
- Make sure all probabilities add up to exactly 1.
- Click Calculate Expectation to generate the result.
- Review the expectation, variance, and standard deviation.
- Study the table and graph for weighted behavior.
- Export the result using the CSV or PDF buttons.
Frequently Asked Questions
1. What is mathematical expectation?
Mathematical expectation is the weighted average of all possible outcomes. It shows the long-run average result you would expect across many repeated trials.
2. When should I use this calculator?
Use it when you have discrete outcomes with known probabilities. It helps in probability, games, finance, risk analysis, and decision-making problems.
3. Why must probabilities sum to 1?
The total probability of all possible outcomes must equal 1 because the distribution must account for every valid event in the model.
4. Can expectation be a value not listed as an outcome?
Yes. The expected value is an average, so it can fall between listed outcomes. That is normal and mathematically correct.
5. What does variance tell me?
Variance measures how spread out the outcomes are around the expected value. Larger variance means greater uncertainty or wider dispersion.
6. What does standard deviation show?
Standard deviation is the square root of variance. It expresses spread in the same unit as the outcomes, making interpretation easier.
7. Can I use decimals for outcomes?
Yes. Outcomes and probabilities can both be decimals. The calculator handles decimal values as long as they are numeric and valid.
8. What do the CSV and PDF exports include?
The exports include the entered data, weighted products, and summary measures such as expected value, variance, and standard deviation.