Expected Value Formula

The expected value of a random variable is the probability-weighted average of all possible outcomes — the long-run mean over many repetitions.

The Formula

E(X)=∑[x×P(x)]E(X) = \sum[x \times P(x)]

When to use: Expected value is what you would "expect" on average after very many trials — not the most likely single outcome, but the center of the distribution.

Quick Example

Fair die: E(X)=(1+2+3+4+5+6)/6=3.5E(X) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5 You never roll 3.5, but it's the average.

Notation

E(X)E(X) or μX\mu_X denotes the expected value of random variable XX

What This Formula Means

The expected value of a random variable is the probability-weighted average of all possible outcomes — the long-run mean over many repetitions.

Expected value is what you would "expect" on average after very many trials — not the most likely single outcome, but the center of the distribution.

Formal View

E(X)=∑ixi P(X=xi)E(X) = \sum_{i} x_i \, P(X = x_i) (discrete); E(X)=∫−∞∞x f(x) dxE(X) = \int_{-\infty}^{\infty} x \, f(x) \, dx (continuous)

Worked Examples

Example 1

easy
A fair six-sided die is rolled. What is the expected value of the outcome?

Answer

E(X)=3.5E(X) = 3.5

First step

1
A fair die has six equally likely outcomes {1,2,3,4,5,6}\{1,2,3,4,5,6\}, each with probability 16\frac{1}{6}.

Full solution

  1. 2
    Apply the expected value formula: E(X)=∑xi⋅P(xi)=16(1+2+3+4+5+6)E(X) = \sum x_i \cdot P(x_i) = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6)
  2. 3
    Compute the sum: 16×21=216=3.5\frac{1}{6} \times 21 = \frac{21}{6} = 3.5
The expected value is the long-run average outcome. Note that 3.53.5 is not a possible outcome of a single roll, but it is the average over many rolls.

Example 2

medium
A game costs $5\$5 to play. You win $20\$20 with probability 0.20.2 and $0\$0 otherwise. What is the expected profit?

Example 3

medium
Flip 3 fair coins. Let XX be the number of heads. Find E(X)E(X).

Common Mistakes

  • Averaging the outcome values without weighting by probability — multiply each value by its chance first.
  • Letting the probabilities not sum to 1 — a complete set of outcomes must have probabilities adding to 1.
  • Treating E(X)E(X) as the result of one trial — it is the long-run average, which need not be an achievable single outcome.

Why This Formula Matters

Expected value turns uncertainty into a single decision number — it is how you judge whether a lottery, insurance policy, or business gamble is worth it, and it generalizes the mean to weighted outcomes. The trap is treating it as a prediction of one trial rather than a long-run average. Recognizing it by "Am I weighting each outcome by its probability and summing to get a long-run average?" — rather than by familiar numbers — is what lets a student tell it apart from mean (of data) and mode / most likely outcome and probability in a mixed problem set.

Frequently Asked Questions

What is the Expected Value formula?

The expected value of a random variable is the probability-weighted average of all possible outcomes — the long-run mean over many repetitions.

How do you use the Expected Value formula?

Expected value is what you would "expect" on average after very many trials — not the most likely single outcome, but the center of the distribution.

What do the symbols mean in the Expected Value formula?

E(X)E(X) or μX\mu_X denotes the expected value of random variable XX

Why is the Expected Value formula important in Math?

Expected value turns uncertainty into a single decision number — it is how you judge whether a lottery, insurance policy, or business gamble is worth it, and it generalizes the mean to weighted outcomes. The trap is treating it as a prediction of one trial rather than a long-run average. Recognizing it by "Am I weighting each outcome by its probability and summing to get a long-run average?" — rather than by familiar numbers — is what lets a student tell it apart from mean (of data) and mode / most likely outcome and probability in a mixed problem set.

What do students get wrong about Expected Value?

The procedure for expected value is the easy part; the trap is averaging the outcome values without weighting by probability. Asking "Am I weighting each outcome by its probability and summing to get a long-run average?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Expected Value formula?

Before studying the Expected Value formula, you should understand: probability, mean.

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