AP Statistics · Unit 2 · 15-25% of the exam · ~24 class periods

Probability, Random Variables, and Probability Distributions

AP Statistics Unit 2 covers probability rules, conditional probability and independence, random variables, the binomial and normal distributions, and an intro to sampling distributions and the central limit theorem. It is 15-25% of the multiple-choice section, about 24 class periods.

AP Statistics: Unit 2 (topics 2.1 Tabular and Graphical Representations for the Distributions of Two Categorical Variables, 2.2 Summary Statistics for Two Categorical Variables, 2.3 Estimating Probabilities Using Simulation, 2.4 Introduction to Probability, 2.5 Mutually Exclusive Events, 2.6 Conditional Probability, 2.7 Independent Events and Unions of Events, 2.8 Introduction to Random Variables and Probability Distributions, 2.9 Parameters of Random Variables, 2.10 The Binomial Distribution, 2.11 The Normal Distribution, 2.12 Sampling Distributions and the Central Limit Theorem). This page maps to the Fall 2026 AP Statistics course, where Unit 2 is 15-25% of the multiple-choice section across about 24 class periods.

What Unit 2 covers

AP Statistics Unit 2, Probability, Random Variables, and Probability Distributions, is 15-25% of the multiple-choice section and takes about 24 class periods. It starts with two-way tables for two categorical variables, builds the rules of probability, and then uses those rules to describe random variables and their probability distributions. The unit ends with the binomial and normal distributions and a first look at sampling distributions and the central limit theorem.

Throughout the unit you write $P(A)$ to mean the probability of event $A$ , a number between 0 and 1 inclusive. Report parameters with units, and connect every probability and parameter to the context of the problem.

The 12 topics in Unit 2

Unit 2 has 12 topics. Here is a faithful summary of each, in the order the College Board lists them.

Topic	What it covers
2.1 Tabular and graphical representations for two categorical variables	Two-way (contingency) tables, side-by-side and segmented bar charts, and mosaic plots used to see whether two categorical variables are associated.
2.2 Summary statistics for two categorical variables	Joint, marginal, and conditional relative frequencies from two-way tables, used to compare distributions and look for association.
2.3 Estimating probabilities using simulation	Random processes, outcomes, and events; probability as a long-run relative frequency; the law of large numbers.
2.4 Introduction to probability	The sample space, equally likely outcomes, a probability from 0 to 1, and the complement rule.
2.5 Mutually exclusive events	Joint probability (the intersection of A and B); two events are mutually exclusive, or disjoint, when they cannot occur together, so their joint probability is 0.
2.6 Conditional probability	The probability of A given B, and the general multiplication rule for the probability that both A and B occur.
2.7 Independent events and unions of events	Independent events, where knowing one does not change the other's probability, and the addition rule for the probability that A or B occurs.
2.8 Introduction to random variables and probability distributions	Discrete random variables, probability distributions whose probabilities add to 1, and cumulative distributions.
2.9 Parameters of random variables	The mean (expected value) and standard deviation of a discrete random variable, interpreted in context.
2.10 The binomial distribution	Binomial conditions, the binomial probability function, the mean, and the standard deviation.
2.11 The normal distribution	The bell-shaped normal curve, its two parameters, the standard normal distribution, the empirical rule, and areas and intervals.
2.12 Sampling distributions and the central limit theorem	The sampling distribution of a statistic, how to simulate it, and the central limit theorem for the mean of a random sample.

Probability rules you will use

Topics 2.3 through 2.7 build the probability rules. A simulation models a random process so that the long-run relative frequency of an event estimates its true probability, and the law of large numbers says that relative frequency settles toward a single value as the number of trials grows.

The sample space is the set of all possible outcomes, and its probability is 1. The complement rule gives the probability that an event does not happen:

P(E^c) = 1 - P(E)

Two events are mutually exclusive (disjoint) when they cannot occur together, so their joint probability $P(A \cap B)$ , the probability of the intersection, is 0. Conditional probability and the addition rule handle events that can overlap:

P(A \mid B) = \frac{P(A \cap B)}{P(B)} \qquad P(A \cup B) = P(A) + P(B) - P(A \cap B)

Events $A$ and $B$ are independent when knowing that one occurred does not change the probability of the other, which means $P(A \cap B) = P(A) \cdot P(B)$ .

Random variables and their parameters

A random variable takes a numerical value produced by a random phenomenon. A probability distribution for a discrete random variable lists every possible value with its probability, and those probabilities add to 1. A cumulative distribution instead gives the probability of a value less than or equal to each outcome.

The mean, also called the expected value, is written $\mu_X$ (mu, the long-run average value of $X$ ). The standard deviation $\sigma_X$ (sigma, the typical distance of the values from the mean) measures spread:

\mu_X = \sum x_i \cdot P(x_i) \qquad \sigma_X = \sqrt{\sum (x_i - \mu_X)^2 \cdot P(x_i)}

Report both with units and in the context of the population they describe. The variance is the square of the standard deviation.

The binomial distribution

A binomial random variable counts the number of successes in a fixed number of independent trials, $n$ , where each trial has two outcomes and a constant probability of success $p$ . The probability of exactly $x$ successes is

P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x}

Its mean is $\mu_X = np$ and its standard deviation is $\sigma_X = \sqrt{np(1 - p)}$ . For the full method and more practice, see binomial probability and the binomial probability calculator.

The normal distribution and the empirical rule

A normal distribution is a continuous, unimodal, bell-shaped, symmetric curve set by two parameters: the mean $\mu$ (mu) and the standard deviation $\sigma$ (sigma). A smaller $\sigma$ makes the curve taller and tighter around the mean, and a larger $\sigma$ spreads it out. The standard normal distribution has $\mu = 0$ and $\sigma = 1$ .

The empirical rule estimates areas under a normal curve: about 68% of values fall within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. This is also called the 68-95-99.7 rule. To place any value on the standard normal scale, use its z-score:

z = \frac{x - \mu}{\sigma}

See the empirical rule (68-95-99.7), how to find a z-score, and the normal distribution calculator.

Sampling distributions and the central limit theorem

A sampling distribution of a statistic is the distribution of that statistic across every possible sample of a given size from a population. You can approximate it by simulation: draw many random samples, record the statistic each time, and graph the results.

The central limit theorem (CLT) states that the sampling distribution of a sample mean $\bar{x}$ (x-bar, the mean of one sample) can be approximated by a normal distribution, and the larger the sample, the better the approximation. Unit 2 introduces this idea, and Units 3 and 4 use it for inference. See the central limit theorem and the sampling distribution and CLT interactive.

Formulas and how to study Unit 2

The AP exam provides a formula sheet and tables for both sections, so you are not memorizing these formulas, but you do decide when to apply each one. The Unit 2 formulas you reach for most are the addition rule, conditional probability, the mean and standard deviation of a discrete random variable, and the binomial probability function. Review them on the formulas page and look up areas on the z-table.

Work each probability problem by writing the rule, substituting the values, and then computing, which is how the rubric awards points. Practice sketching, shading, and labeling a normal curve before you calculate an area. For the full course map, see AP Statistics, then move on to Unit 3: Inference for Proportions.

Reading a two-way table: conditional probability and the addition rule

A survey of 100 students records whether each owns an e-reader and whether each read a book in the past month.

	Read: yes	Read: no	Total
E-reader: yes	24	6	30
E-reader: no	42	28	70
Total	66	34	100

Find the probability that a student read a book given that they own an e-reader, and the probability that a student owns an e-reader or read a book.

Conditional probability restricts to the e-reader owners. There are 30 owners and 24 of them read a book, so $P(\text{read} \mid \text{e-reader}) = \frac{24}{30} = 0.80$ .
Check with the formula. $P(\text{read} \cap \text{e-reader}) = \frac{24}{100} = 0.24$ and $P(\text{e-reader}) = \frac{30}{100} = 0.30$ , so $\frac{0.24}{0.30} = 0.80$ .
For the union, use the addition rule. $P(\text{e-reader}) = 0.30$ , $P(\text{read}) = \frac{66}{100} = 0.66$ , and $P(\text{read} \cap \text{e-reader}) = 0.24$ .
$P(\text{e-reader} \cup \text{read}) = 0.30 + 0.66 - 0.24 = 0.72$ . Check by counting the students in at least one category: $24 + 6 + 42 = 72$ out of 100.

$P(\text{read} \mid \text{e-reader}) = 0.80$ and $P(\text{e-reader} \cup \text{read}) = 0.72$ .

Mean and standard deviation of a discrete random variable

Let $X$ be the number of pets a randomly chosen household owns, with this probability distribution. Find the mean and standard deviation of $X$ .

Pets $x$	0	1	2	3
$P(x)$	0.3	0.4	0.2	0.1

Confirm the probabilities add to 1: $0.3 + 0.4 + 0.2 + 0.1 = 1$ .
Mean (expected value): $\mu_X = \sum x_i P(x_i) = 0(0.3) + 1(0.4) + 2(0.2) + 3(0.1) = 0 + 0.4 + 0.4 + 0.3 = 1.1$ pets.
Squared deviations times probability: $(0 - 1.1)^2(0.3) = 1.21(0.3) = 0.363$ ; $(1 - 1.1)^2(0.4) = 0.01(0.4) = 0.004$ ; $(2 - 1.1)^2(0.2) = 0.81(0.2) = 0.162$ ; $(3 - 1.1)^2(0.1) = 3.61(0.1) = 0.361$ .
Variance: $0.363 + 0.004 + 0.162 + 0.361 = 0.890$ . Standard deviation: $\sigma_X = \sqrt{0.890} \approx 0.943$ pets.

$\mu_X = 1.1$ pets and $\sigma_X \approx 0.943$ pets.

Binomial probability, mean, and standard deviation

A 10-question multiple-choice quiz has 4 options per question. A student guesses every answer, so the probability of a correct guess is $p = 0.25$ . Let $X$ be the number of correct answers. Find the mean, the standard deviation, and the probability of exactly 3 correct.

This is binomial with $n = 10$ and $p = 0.25$ because there are 10 independent trials, two outcomes per question, and a constant success probability.
Mean: $\mu_X = np = 10(0.25) = 2.5$ questions.
Standard deviation: $\sigma_X = \sqrt{np(1 - p)} = \sqrt{10(0.25)(0.75)} = \sqrt{1.875} \approx 1.369$ questions.
Exactly 3 correct: $\binom{10}{3} = 120$ , $0.25^3 = 0.015625$ , and $0.75^7 \approx 0.13348$ .
$P(X = 3) = 120(0.015625)(0.13348) = 1.875(0.13348) \approx 0.2503$ .

$\mu_X = 2.5$ , $\sigma_X \approx 1.369$ , and $P(X = 3) \approx 0.2503$ .

Empirical rule and z-scores on a normal distribution

A large set of exam scores is approximately normal with mean $\mu = 500$ and standard deviation $\sigma = 100$ . Use the empirical rule to find the percent of scores between 300 and 700 and the percent above 700, then find the z-score of a 650.

The bounds 300 and 700 are each 2 standard deviations from the mean: $500 - 2(100) = 300$ and $500 + 2(100) = 700$ .
By the empirical rule, about 95% of values fall within 2 standard deviations of the mean, so about 95% of scores are between 300 and 700.
The remaining 5% is split between the two tails, so about $\frac{5\%}{2} = 2.5\%$ of scores are above 700.
z-score of 650: $z = \frac{x - \mu}{\sigma} = \frac{650 - 500}{100} = 1.5$ , so 650 sits 1.5 standard deviations above the mean.

About 95% of scores fall between 300 and 700, about 2.5% are above 700, and a 650 has a z-score of 1.5.

Frequently asked questions

How much of the AP Statistics exam is Unit 2?

Unit 2 is 15-25% of the multiple-choice section, and the College Board plans about 24 class periods for it. Its ideas also appear inside free-response questions, since probability and distributions support the inference in later units.

What is the difference between mutually exclusive and independent events?

Mutually exclusive (disjoint) events cannot happen at the same time, so their joint probability is 0. Independent events can happen together, but knowing that one occurred does not change the probability of the other. Two events that both have nonzero probability cannot be mutually exclusive and independent at once.

Do I have to memorize the binomial and normal formulas?

No. The AP exam gives you a formula sheet and tables for both sections, including the binomial probability function and its mean and standard deviation. Your job is to recognize when a setting is binomial or normal and to apply the right formula with the values from the problem.