Expected Value Calculator

What Is Expected Value?

Expected value $E(X)$ is the long-run average of a discrete random variable. Multiply each outcome by its probability and sum the products. If you repeated the experiment many times, the sample mean would approach $E(X)$. For probability basics, see the Probability Calculator or Dice Average Calculator.

Expected Value Formula

For outcomes $x_1, x_2, \ldots, x_n$ with probabilities $P(x_1), P(x_2), \ldots, P(x_n)$:

$$E(X) = \sum_{i=1}^{n} x_i \cdot P(x_i)$$

Example: Fair Die Roll

Each face 1 through 6 has probability $1/6$. Then $E(X) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + \cdots + 6 \cdot \frac{1}{6} = 3.5$. A single roll cannot show 3.5, but the average over many rolls approaches that value.

Example: Should You Take a Bet?

Win $100 with probability 0.35, lose $45 with probability 0.65. $E(X) = 100(0.35) + (-45)(0.65) = 5.75$. Positive expected value suggests the bet is favorable over many repetitions.

Frequently Asked Questions

Can expected value be negative?

Yes. If losses outweigh gains on average, $E(X)$ is negative. That means you expect to lose money over many trials.

Must probabilities sum to exactly 1?

Yes. All outcomes must be covered and probabilities must total 1 for a valid probability distribution.

How is this related to weighted average?

Expected value is a weighted average where the weights are probabilities instead of arbitrary weights.

How many outcomes can I enter?

Up to 20 outcome-probability pairs. Add rows as needed and remove extras you do not use.

Is this the same as chi-square expected counts?

Related idea but different context. Chi-square expected cell counts use row and column totals; this tool computes $E(X)$ for a discrete random variable.

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