AP Statistics, AP exams

Unit 2: Probability, Random Variables, and Probability Distributions

Unit 2 supplies the probability machinery that inference depends on. It covers probability rules, conditional probability and independence, random variables and their distributions, and the binomial and normal models. The central limit theorem and the behavior of sampling distributions bridge from probability to the inference units that follow.

Position: 2 of 5 units
Topics covered: 6 topics in this guide

What is covered

Topics inside this unit.

These topics describe the scope students most often review for this part of AP Statistics. Treat them as a study checklist rather than an exhaustive list of what a single exam form can sample.

Probability rules: Addition and multiplication rules, complements, and simulation as a way to estimate a probability.
Conditional probability and independence: Computing conditional probabilities from two-way tables, and testing whether events are independent.
Random variables: Discrete and continuous random variables, expected value, and standard deviation.
Binomial distributions: The conditions that define a binomial setting, and computing probabilities, mean, and standard deviation.
Normal distributions: Using the normal model for probabilities and percentiles, and checking whether it is appropriate.
Sampling distributions and the central limit theorem: The distribution of a statistic across repeated samples, and why sample means become approximately normal.

Study tips

How to practice Probability.

Section-specific guidance that follows from how this material is usually tested.

Check the binomial conditions explicitly before using the model; the exam awards points for the check.
Distinguish the distribution of a population, of a sample, and of a statistic; conflating them causes most inference errors later.
Practice simulation reasoning, since the exam asks students to design and interpret simulations as well as compute exactly.

Where this fits

Unit 2: Probability, Random Variables, and Probability Distributions is 2 of 5 units in the AP Statistics guide. Use the full guide to balance it against everything else you have to review.

Back to the AP Statistics guide Official exam information

Sample Probability questions

Practice questions, not yet expert-reviewed

Original practice questions written to the published framework for this section. No expert has reviewed them, and they are not taken from any official exam. Try each one before you open the answer.

Question 1Foundational
Events A and B are mutually exclusive, with P(A) = 0.30 and P(B) = 0.45. What is P(A or B)?
1. A0.75
2. B0.615
3. C0.135
4. D0.15
Show answer
Answer: A. 0.75
Mutually exclusive events cannot happen together, so P(A and B) = 0 and the addition rule reduces to P(A) + P(B) = 0.75. The value 0.615 subtracts the product 0.135, which would be correct only if the events were independent rather than mutually exclusive.
Key insight: Mutually exclusive means the overlap is zero; independent means the overlap equals the product.
Question 2Foundational
Among 200 students, 120 play a sport, and 45 of those 120 also play an instrument. If a student who plays a sport is chosen at random, what is the probability that the student plays an instrument?
1. A0.375
2. B0.225
3. C0.6
4. D0.27
Show answer
Answer: A. 0.375
The condition restricts the sample space to the 120 sport players, so the probability is 45/120 = 0.375. Dividing 45 by the full 200 gives 0.225, which answers a different question about all students rather than about sport players only.
Key insight: Conditional probability changes the denominator to the size of the group you conditioned on.
Question 3Intermediate
A binomial random variable counts successes in n = 10 independent trials with probability of success p = 0.4. What is its mean?
1. A4
2. B2.4
3. C6
4. D0.4
Show answer
Answer: A. 4
The mean of a binomial distribution is np = 10(0.4) = 4 successes. The value 2.4 is the variance np(1 - p), whose square root is the standard deviation of about 1.55, so it answers a different question.
Key insight: Binomial mean is np; binomial variance is np(1 - p) and standard deviation is its square root.
Question 4Intermediate
A game costs $2 to play. A player wins $10 with probability 0.1 and wins nothing otherwise. What is the expected net gain per play?
1. A-$1.00
2. B$1.00
3. C-$0.80
4. D$8.00
Show answer
Answer: A. -$1.00
The expected winnings are 0.1($10) + 0.9($0) = $1.00, and subtracting the $2 cost gives an expected net gain of -$1.00 per play. Reporting $1.00 forgets to subtract the cost of playing.
Key insight: Expected value multiplies each outcome by its probability, then accounts for any fixed cost.
Question 5Advanced
A population has mean 60 and standard deviation 18. For random samples of size 36, what is the standard deviation of the sampling distribution of the sample mean?
1. A3
2. B18
3. C6
4. D0.5
Show answer
Answer: A. 3
The standard deviation of the sample mean is σ/√n = 18/√36 = 18/6 = 3. Reporting 18 uses the population standard deviation and ignores that averaging reduces variability, and 0.5 divides by n = 36 instead of by √n = 6.
Key insight: Averaging shrinks variability by a factor of √n, which is why larger samples give more precise estimates.

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Unit 2: Probability, Random Variables, and Probability Distributions