Expected Value Formula

The Formula

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 = 3.5 You never roll 3.5, but it's the average.

Notation

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

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) = \sum_{i} x_i \, P(X = x_i) (discrete); E(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?

Solution

  1. 1
    A fair die has six equally likely outcomes \{1,2,3,4,5,6\}, each with probability \frac{1}{6}.
  2. 2
    Apply the expected value formula: E(X) = \sum x_i \cdot P(x_i) = \frac{1}{6}(1 + 2 + 3 + 4 + 5 + 6)
  3. 3
    Compute the sum: \frac{1}{6} \times 21 = \frac{21}{6} = 3.5

Answer

E(X) = 3.5
The expected value is the long-run average outcome. Note that 3.5 is not a possible outcome of a single roll, but it is the average over many rolls.

Example 2

medium
A game costs \5 to play. You win \20 with probability 0.2 and \0$ otherwise. What is the expected profit?

Common Mistakes

  • Forgetting to weight each outcome by its probability — simply averaging the possible values without accounting for their likelihood
  • Expecting the expected value to occur on a single trial — E(X) = 3.5 for a die does not mean you will ever roll 3.5
  • Adding probabilities instead of multiplying each outcome by its probability before summing

Why This Formula Matters

Basis for decision-making under uncertainty, insurance, gambling.

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) or \mu_X denotes the expected value of random variable X

Why is the Expected Value formula important in Math?

Basis for decision-making under uncertainty, insurance, gambling.

What do students get wrong about Expected Value?

A game is 'fair' when expected value = 0 (break even long-term).

What should I learn before the Expected Value formula?

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

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Related Concepts

ProbabilityMeanVariance