Unit 4: Probability, Random Variables, and Probability Distributions
AP Statistics: 52 practice questions with detailed explanations.
Unit Study Guide
Executive Summary
Probability formalizes chance. Random variables attach numbers to outcomes, and their distributions give means, variances, and the probabilities of interest.
Basic probability
P(A) is between 0 and 1; all outcomes sum to 1. Addition rule: P(A or B) = P(A) + P(B) - P(A and B); disjoint events drop the overlap. Complement: P(not A) = 1 - P(A). Conditional: P(A | B) = P(A and B) / P(B). Independence: P(A and B) = P(A)·P(B), or P(A | B) = P(A).
Discrete random variables
A random variable assigns a number to each outcome. Expected value (mean) = sum of value × probability. Variance measures expected squared deviation; standard deviation is its square root. Adding a constant shifts the mean only; multiplying scales both mean and spread.
Binomial
Counts successes in n independent trials with constant p. Mean = np; variance = np(1-p). P(X = k) uses the binomial coefficient. Use complements for at-least problems.
Geometric
Counts trials until the first success. Mean = 1/p. P(first success on trial k) = (1-p)^(k-1) · p.
Continuous distributions
Values fill intervals; P(exact value) = 0; probabilities are areas under a density curve; total area = 1. Normal probabilities use z-scores and the empirical rule.
Exam traps
Disjoint is not independent — mutually exclusive events are dependent. Multiply for AND only when independent. Binomial counts successes; geometric counts trials.