Random variables and expected value

Turn chance outcomes into a number, then find its long-run average.

Learning goals

Build a probability distribution for a discrete random variable.
Compute and interpret expected value in context.

Explanation

A random variable assigns a number to each outcome. For a discrete variable, list the possible values and their probabilities. Those probabilities must be between 0 and 1 and add to 1.

Expected value is a weighted average of those values. It describes what happens if the chance process is repeated many times. A single trial can land far from the expected value, especially when the distribution is spread out.

Key terms

Random variable. A numerical outcome of a chance process.
Expected value. The long-run average value of a random variable, found by weighting each outcome by its probability.

Common mistakes

Calling the expected value the most likely single outcome.
Using a binomial model when trials are not independent or the success probability changes.

Practice

Original Marlow Works items. Check the answer explanation after you try.

1 of 1Probability, Random Variables, and Probability Distributions · medium · short answer

A fair spinner has four equal sectors labeled A, B, C, and D. What is P(A or B)? Show brief reasoning.

Original Marlow Works item — not a College Board question.

Your response

Take your time—this is practice, not a test.

Answer explanation

A spinner pays 0, 2, or 10 points with probabilities 0.5, 0.4, and 0.1. What is the expected score?

Answer. 0(0.5) + 2(0.4) + 10(0.1) = 1.8 points.

Expected value weights each score by how often it should appear in the long run.

Previous lesson Course practice

Related resources

External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.

CC BY — attribution required
Collaborative Statistics
Open statistics text for distributions, probability, and inference.
Open Textbook Library listing · CC BY · accessed 2026-10-01
Open source