Expected Value Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Expected Value.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

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.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: E[X] = \sum x_i P(x_i): multiply each outcome by its probability and sum. Expected value is linear: E[aX + b] = aE[X] + b.

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

Sense of Study hint: Make a two-column table: each possible outcome and its probability. Multiply across each row, then add all the products.

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?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium
A raffle has 100 tickets. One ticket wins \500 and two tickets win \50 each. Each ticket costs \10$. Find the expected net gain per ticket.

Example 2

medium
A prize wheel pays \0, \2, \5, and \20 with probabilities 0.50, 0.30, 0.15, and 0.05, respectively. What is the expected payout per spin?
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Related Concepts

ProbabilityMeanVariance

Background Knowledge

These ideas may be useful before you work through the harder examples.

probabilitymean