AP Statistics — probability and sampling distributions

Probability and sampling distributions (Unit 2 of the current course framework, and the opening topics of Units 3 and 4) are where AP Statistics stops describing data and starts modeling it, and they are where most of the course's arithmetic lives. Ten questions here, one per idea. The probability half opens with the addition rule asked so that the overlap is the whole point — ten percent of an invented town owns both a bicycle and a scooter, and those residents have already been counted twice. Independence gets its own question (it closes the set), not as a formula to recall but as a judgment to make: half a school studies a language and half the orchestra does too, so orchestra membership tells you nothing, and the wrong answers are the three ways that comparison is usually botched. The multiplication rule is asked without replacement, where the second draw is conditional on the first and treating the two as independent gives a visibly different number.

Random variables come next. Expected value is asked with a distribution whose mean, mode and unweighted average are three different numbers, so that a candidate who ignores the probabilities, or who names the commonest outcome, or who insists the answer must be a score the spinner can actually award, each lands somewhere distinct. The binomial standard deviation turns on a square root that catches everyone: a hundred inspections with a 20 percent defect rate give a mean of 20 and a standard deviation of 4, not 16. The binomial setting is tested by elimination — a varying number of trials, a probability that shifts as cards leave the deck, and a measurement rather than a success or failure — and a binomial probability of at least one success among three guesses, found through the complement.

Three questions build to the sampling distribution. A z-score is used for what it is for, comparing two scores from distributions with different centers and different spreads. The central limit theorem is asked against a strongly skewed population, because the point of the theorem is the non-normal case, and because the three commonest misreadings — that the population becomes normal, that one sample becomes normal, that the theorem needs normality to start with — are all worth meeting once. The third asks what quadrupling a sample does to the variability of its mean: it halves it, since the square root of four is two, and four times the data buys twice the precision rather than four times.

Every town, factory, school and test in the bank is invented, and the questions say so. Nothing is drawn from any College Board publication, released examination, course and exam description, scoring guideline, formula sheet or normal table. There are no tables; a few items and cards carry a drawing, as on the exam — Venn diagrams, normal curves and a tree diagram — but every distribution is described in words, so most of the display-reading that some real probability questions require is not practiced here.

Zestly is an independent study tool. It is not affiliated with the College Board, which owns the AP Statistics exam, and it is not a test center.

• Apply the addition rule to overlapping events, subtracting the overlap exactly once
• Decide whether two characteristics are independent by comparing a conditional proportion with its unconditional counterpart
• Compute a probability for two draws without replacement, adjusting the second for the first
• Calculate an expected value and read it as a long-run average that need not be an attainable outcome
• Calculate the mean and standard deviation of a binomial random variable and interpret them in context
• Test a situation against all four conditions of the binomial setting
• Compute a binomial probability, using the complement for 'at least one'
• Use a z-score to compare values drawn from distributions with different means and spreads
• State what the central limit theorem describes, and what it does not say about the population or about one sample
• Work out how the variability of a sample mean changes with the sample size, through the square root

Practice material written by Zestly, based on the published AP Statistics course framework (AP Statistics Course and Exam Description, effective fall 2026, College Board). Every data set, context and number is invented; no official exam material is reproduced.