What it covers: Probability rules, conditional probability and independence, random variables, and the binomial, normal, and sampling distributions.
Exam weight: About 15–25% of the AP Statistics exam.
The big question: How do we quantify chance, and how do random variables and their distributions model uncertain outcomes?
Statistical practices: Formulate Questions (P1), Collect Data (P2), Analyze Data (P3), Interpret Results (P4).
Key topics at a glance
Two-Way Tables
Marginal = row/column total ÷ grand total. Joint = one cell ÷ grand total. Conditional = cell ÷ row or column total. Differing conditional distributions ⇒ association.
Probability Basics
Probability is long-run relative frequency (Law of Large Numbers). Outcomes in a sample space are between 0 and 1 and sum to 1. Simulation estimates hard probabilities.
Addition Rules
General: P(A or B) = P(A) + P(B) − P(A and B). Mutually exclusive (disjoint): P(A and B) = 0, so P(A or B) = P(A) + P(B).
Conditional & Independence
P(A | B) = P(A and B) / P(B). Independent if P(A | B) = P(A), equivalently P(A and B) = P(A)·P(B).
Random Variables
A probability distribution lists values and probabilities (sum to 1). Expected value E(X) = Σ x·P(x); standard deviation measures spread of outcomes.
Combining Variables
Means add: E(X + Y) = E(X) + E(Y). Variances add for independent variables: Var(X ± Y) = Var(X) + Var(Y). Standard deviations do not add.
Binomial Distribution
BINS: Binary, Independent, fixed Number n, Same p. Mean = np, SD = √(np(1 − p)). P(X = k) uses C(n, k)·p^k·(1 − p)^(n − k).
Normal & Sampling Distributions
z = (x − mean)/SD. Empirical rule: 68–95–99.7. The Central Limit Theorem: for large n the sampling distribution of the mean is approximately normal.
The key terms you must know
Conditional relative frequency — a two-way-table cell divided by its row or column total.
Association — present when the conditional distributions differ across categories.
Law of Large Numbers — the observed proportion approaches the true probability as trials increase.
Mutually exclusive (disjoint) — events that cannot both occur; P(A and B) = 0.
General addition rule — P(A or B) = P(A) + P(B) − P(A and B).
Conditional probability — P(A | B) = P(A and B) / P(B).
Independent events — P(A | B) = P(A); P(A and B) = P(A)·P(B).
Random variable — assigns a number to each outcome of a chance process.
Expected value — the mean of a random variable, Σ x·P(x).
Binomial setting (BINS) — Binary, Independent, fixed Number, Same probability.
Normal distribution — a symmetric bell curve set by its mean and standard deviation.
z-score — (x − mean)/SD, the number of standard deviations from the mean.
Central Limit Theorem — the sampling distribution of the mean is approximately normal for large n.
Key themes to remember
Restrict the sample space for conditional probability. P(A | B) only counts outcomes where B happens.
Independence is about information. Two events are independent when knowing one tells you nothing about the other.
Distributions have a center and a spread. Expected value and standard deviation summarize a random variable just like mean and SD summarize data.
Variances add, standard deviations don't. When combining independent random variables, add the variances, then take the square root.
Binomial and normal are the two key models. Recognize the setting, then use its formulas — and the CLT links them to inference.
Common exam traps
Mutually exclusive ≠ independent. Disjoint events are actually dependent — if one happens, the other cannot.
Don't forget to subtract the overlap. Use the general addition rule unless the events are truly mutually exclusive.
Check all four BINS conditions before calling something binomial; a changing probability or non-fixed n breaks it.
SD is not np(1 − p). The binomial standard deviation is the square root: √(np(1 − p)).
Add variances, not standard deviations, when combining independent random variables.
The empirical rule needs a normal shape. 68–95–99.7 does not apply to skewed distributions.
A z-score is not a probability. Convert the z-score to a probability using the table or technology.