Expected Value Calculator
An Expected Value Calculator helps you find the average outcome you can expect from a probability distribution. Expected value is widely used in probability, statistics, finance, economics, games, and decision-making.
What Is Expected Value?
Expected value is the weighted average of all possible outcomes of a random variable. Each outcome is multiplied by its probability, and the results are then added together.
The basic formula is:
E(X) = Σ [x × P(x)]
Where x represents a possible outcome and P(x) represents the probability of that outcome.
How to Use the Expected Value Calculator
- Enter each possible outcome.
- Enter the probability associated with each outcome.
- Make sure the probabilities represent the complete set of possible outcomes.
- Click the calculate button.
- The calculator will determine the expected value.
Expected Value Example
Suppose a game has two possible outcomes. You can win $10 with a probability of 0.4 or win $20 with a probability of 0.6.
The expected value is:
E(X) = (10 × 0.4) + (20 × 0.6)
E(X) = 4 + 12 = 16
Therefore, the expected value is $16.
Expected Value for a Probability Distribution
For multiple possible outcomes, multiply every outcome by its probability and add the products together.
| Outcome | Probability | Outcome × Probability |
|---|---|---|
| 5 | 0.2 | 1 |
| 10 | 0.5 | 5 |
| 20 | 0.3 | 6 |
The expected value is:
E(X) = 1 + 5 + 6 = 12
Why Is Expected Value Important?
Expected value provides a useful way to summarize the long-term average outcome of a random process. It can help compare different choices when their possible outcomes and probabilities are known.
Expected value is commonly used in:
- Probability and statistics
- Financial analysis
- Economics
- Risk assessment
- Games and probability problems
- Business decision-making
Important Note About Expected Value
The expected value does not necessarily represent an outcome that will actually occur in a single trial. Instead, it describes the theoretical average result over many repeated trials under the same probability conditions.
Frequently Asked Questions
What is the formula for expected value?
For a discrete random variable, the formula is E(X) = Σ [x × P(x)].
Do probabilities have to add up to 1?
For a complete probability distribution, the probabilities of all possible outcomes should add up to 1, or 100%.
Can expected value be negative?
Yes. If the weighted average of the possible outcomes is negative, the expected value will also be negative.
Is expected value the same as average?
Expected value is a probability-weighted average. Unlike an ordinary average, outcomes with higher probabilities have a greater influence on the result.
Related Calculators
For more statistical calculations, explore our Variance Calculator and Standard Deviation Calculator. You can also visit our Math Calculators for more useful mathematical tools.
Conclusion
The Expected Value Calculator makes it easy to calculate the probability-weighted average of possible outcomes. It is a useful tool for understanding probability distributions and making informed comparisons when different outcomes have different probabilities.