Review AP Statistics Unit 2 to build the probability foundation that powers every inference procedure in the course. From two-way tables and simulation to binomial distributions, normal models, and the Central Limit Theorem, this unit connects chance to data.
Use the topic guides, key terms, and practice questions available here to work through each concept before your exam.
Unit 2 is the probability core of AP Statistics. It begins by extending data analysis to two categorical variables using two-way tables and bar charts, then builds a complete probability toolkit: simulation, complement and addition rules, conditional probability, independence, and mutually exclusive events. The second half shifts to random variables, covering discrete distributions, expected value, standard deviation, the binomial model, the normal model, and finally sampling distributions and the Central Limit Theorem.
Two-way tables organize counts or relative frequencies for two categorical variables. Joint, marginal, and conditional relative frequencies let you compare distributions and evaluate association. Side-by-side bar charts, segmented bar charts, and mosaic plots display the same information visually.
Probability is the long-run relative frequency of an event. Core rules include the complement rule, addition rule, general multiplication rule, and conditional probability formula P(A|B) = P(A and B) / P(B). Discrete random variables assign numbers to outcomes; their distributions list every value and its probability, and parameters include the expected value and standard deviation.
The binomial distribution models the count of successes in n independent trials with fixed probability p. The normal distribution is a continuous, symmetric, bell-shaped model described by mean and standard deviation. The Central Limit Theorem explains why sample means follow an approximately normal sampling distribution for large enough sample sizes, regardless of the population shape.
Every confidence interval and hypothesis test in Units 3 through 5 asks: how likely is this sample result if the null hypothesis is true? Answering that question requires the probability rules, distribution models, and sampling distribution logic built in Unit 2. Without this unit, inference is just arithmetic with no meaning.
Build and interpret two-way tables, side-by-side bar charts, segmented bar charts, and mosaic plots to compare distributions of two categorical variables.
Calculate joint, marginal, and conditional relative frequencies from two-way tables and use them to evaluate association between two categorical variables.
Design simulations to estimate probabilities, interpret relative frequency as an estimate of true probability, and apply the Law of Large Numbers.
Define sample spaces, assign probabilities using equally likely outcomes, and apply the complement rule P(not E) = 1 - P(E).
Identify disjoint events using joint probability, explain why P(A and B) = 0 for mutually exclusive events, and distinguish this from independence.
Calculate P(A|B) using the formula P(A and B) / P(B), apply the general multiplication rule, and use two-way tables or tree diagrams to organize conditional probability problems.
Test independence using conditional probability, apply the multiplication rule for independent events, and use the addition rule P(A or B) = P(A) + P(B) - P(A and B).
Define discrete random variables, construct valid probability distributions as tables or graphs, and read cumulative probability distributions.
Calculate and interpret the expected value mu_X and standard deviation sigma_X of a discrete random variable using the weighted sum formulas.
Verify BINS conditions, calculate binomial probabilities using C(n,x) p^x (1-p)^(n-x), and find the mean np and standard deviation sqrt(np(1-p)).
Apply the Empirical Rule, convert values to z-scores, find probabilities as areas under the normal curve, and determine percentile cutoffs using inverse normal calculations.
Describe sampling distributions through simulation, state and apply the CLT for sample means, and explain how sample size affects the shape and spread of a sampling distribution.
Open this guide for a closer review of the topic.
A two-way table (contingency table) cross-classifies two categorical variables. Cells hold counts or relative frequencies. Three types of relative frequency matter for comparing variables and justifying claims about association.
| Frequency type | Denominator | What it describes |
|---|---|---|
| Joint | Table total | Proportion in one specific cell |
| Marginal | Table total | Overall distribution of one variable |
| Conditional (row) | Row total | Distribution of column variable within one row category |
| Conditional (column) | Column total | Distribution of row variable within one column category |
Simulation estimates probability by running a random process many times and recording relative frequencies. Formal probability assigns numbers to events using a sample space and fixed rules.
| Approach | Source | When to use |
|---|---|---|
| Simulation / empirical | Observed trials or technology | True probability unknown or complex |
| Theoretical | Sample space structure | Equally likely outcomes, simple events |
These three topics give you the full toolkit for combining and conditioning probabilities. Confusing mutually exclusive with independent is the most common error in this section.
| Property | Definition | Key formula |
|---|---|---|
| Mutually exclusive | Cannot occur together | P(A and B) = 0 |
| Independent | Knowing one does not change the other | P(A and B) = P(A) x P(B) |
| Neither | Can occur together and are related | Use P(A|B) = P(A and B) / P(B) |
A discrete random variable X assigns a numerical value to each outcome of a random process. Its probability distribution lists every possible value and the probability of each. Parameters describe the center and spread of that distribution.
The binomial model applies when a fixed number of independent trials each have exactly two outcomes (success or failure) with the same probability of success. The acronym BINS summarizes the four conditions: Binary outcomes, Independent trials, fixed Number of trials, Same probability of success.
The normal distribution is a continuous, symmetric, bell-shaped model fully described by its mean mu and standard deviation sigma. Probability equals area under the curve. You find areas using z-scores with a standard normal table or technology.
A sampling distribution describes how a sample statistic varies across all possible samples of size n from a population. The Central Limit Theorem (CLT) states that for large enough n, the sampling distribution of the sample mean is approximately normal regardless of the population shape.
| Term | Definition |
|---|---|
| Law of Large Numbers | As the number of independent trials increases, the observed relative frequency of an outcome converges to its true probability. |
| Empirical Rule | For a normal distribution, approximately 68% of values fall within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3. Also called the 68-95-99.7 rule. |
| Simpson's Paradox | A trend seen in aggregated data reverses or disappears when the data are broken into subgroups; a reminder to examine conditional distributions rather than only marginal totals. |
| Bayes' theorem | A formula for updating a conditional probability with new information: P(A|B) = P(B|A) × P(A) / P(B), applied when reversing the direction of conditioning. |
| law of total probability | The total probability of an event equals the sum of its probabilities across each partition of the sample space, weighted by the probability of each partition. |
Mutually exclusive events have P(A and B) = 0, which actually means they are dependent (knowing A occurred tells you B cannot occur). Independent events can both happen; they just do not influence each other's probability. These two properties are almost never both true at the same time for events with positive probability.
P(A|B) divides by P(B), not by the table total. When working from a two-way table, restrict your denominator to the row or column total for the given condition, not the grand total.
The binomial formula only applies when all four conditions hold: binary outcomes, independent trials, fixed n, and constant p. Sampling without replacement from a small population violates independence, so the binomial is only approximately valid when the sample is less than 10% of the population.
The normal model is a continuous approximation. When a problem asks for the probability of an exact value from a continuous distribution, the answer is 0. Probabilities correspond to intervals, not single points.
The expected value mu_X is the long-run average over many repetitions, not a prediction for any single trial. For a binomial with n = 10 and p = 0.3, the expected value is 3, but 3 successes is not guaranteed on any given set of 10 trials.
Free-response questions frequently present a two-way table and ask you to determine whether two categorical variables are associated. The expected task is to compute and compare conditional relative frequencies across groups, then write a conclusion in context. Saying the distributions look different is not enough; you must reference specific numerical evidence from the conditional distributions.
Multiple-choice and free-response items test whether you can select and correctly apply the right probability rule. Common task patterns include identifying whether events are independent or mutually exclusive, setting up a conditional probability from a two-way table, and using the addition rule without double-counting. Writing probability notation such as P(A|B) and showing the formula before substituting numbers earns credit on free-response items.
The binomial, normal, and sampling distribution topics in Unit 2 reappear directly in Units 3 and 4 as the basis for confidence intervals and hypothesis tests. Exam questions may ask you to verify conditions for using a normal approximation, interpret a standard error, or explain what the Central Limit Theorem guarantees about the shape of a sampling distribution. Precise language about shape, center, and spread of a sampling distribution is expected.
Open the individual guides for Unit 2 when you want a closer review of one topic.
browse guidesUse AP-style practice after you review the notes so you can check what you understand.
start practicePractice free-response reasoning and compare your answer with scoring guidance.
practice FRQsWatch past review streams filtered to Unit 2 when you want a video walkthrough.
open videosOpen the Fiveable one-page unit review, then explore visual cheatsheets for a quick refresher.
open unit cheatsheetEstimate your broader AP score goal after you review the course and exam format.
open calculatorAP Stats Unit 2, Probability, Random Variables, and Probability Distributions, contains 12 topics covering two-way tables; simulation; probability rules; mutually exclusive, conditional, and independent events; discrete random variables; means and standard deviations of random variables; binomial distributions; normal distributions; sampling distributions; and the central limit theorem. These are the Fall 2026 CED topics used for the May 2027 exam.
Unit 2 accounts for 15–25% of the multiple-choice section. That range is a section weight, not a promise that every practice form will use the same percentage, so review the full unit rather than trying to predict one exact question count.
By the end of Unit 2, you should be able to translate a context into events, use probability rules without mixing up conditional and joint probability, calculate binomial or normal probabilities, and describe the shape, center, and variability of a sampling distribution. The exam rewards the method and the interpretation, so practice writing what a result means in the problem's context instead of stopping at a calculator output.
Common Unit 2 mistakes include assuming mutually exclusive events are independent, reversing a conditional probability, using a binomial model without checking its conditions, or forgetting that a sampling-distribution standard deviation depends on sample size. Slow down long enough to identify the variables, population, and requested conclusion before calculating.
Start with the vocabulary and conditions, then mix short calculations with full-sentence interpretations. For each missed question, label whether the problem was choosing a method, doing the calculation, or interpreting the result. Finish with timed mixed practice so you have to recognize the method without a topic label.
Sampling distributions and the central limit theorem provide the probability foundation for the confidence intervals and hypothesis tests in Units 3 and 4.