Expected Value Calculator
Enter a set of values and their respective probabilities to calculate the expected value.
Results: Expected Value
Expected Value Formula:
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How to Calculate the Expected Value
Expected value is a statistical measure that represents the average outcome you can expect from a random event if the same experiment is repeated many times. Rather than predicting what will happen in a single trial, expected value estimates what the average outcome will be over many trials.
To calculate the expected value, multiply each possible outcome by the probability that it will occur, then add all of the products together. Outcomes with higher probabilities contribute more to the final result than those that are less likely to occur.
Expected Value Formula
The expected value formula is:
Thus, the expected value E(X) is equal to the sum of the products of each value x and its corresponding probability P(x).
The expected value formula can also be written as:
E(X) = x1 · P(x1) + ··· + xn · P(xn)
To calculate the expected value using the formula, start by listing each possible outcome and its probability. Then, multiply each outcome by its probability. And finally, add each of these products together.
For example, let’s calculate the expected value of a game that has the following possible outcomes:
| Outcome | Probability |
|---|---|
| $10 | 0.2 |
| $25 | 0.5 |
| $50 | 0.3 |
Calculate the product of each value and its probability:
10 · 0.2 = 2
25 · 0.5 = 12.5
50 · 0.3 = 15
Then, add these together to calculate the expected value:
E(X) = 2 + 12.5 + 15 = 29.5
Therefore, the expected value is $29.50.
Expected Value vs. Weighted Average
Expected value is mathematically equivalent to a weighted average when the weights are probabilities.
In a standard average, every value contributes equally because each value has the same weight. In a weighted average, some values have more influence than others because they are assigned different weights.
Expected value uses probabilities as the weights. Outcomes that are more likely to occur have a greater effect on the final result than less likely outcomes.
Because the probabilities always sum to 1, the expected value is simply the probability-weighted average of all possible outcomes.