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Probability, random variables and probability distributions practice problems AP Statistics, Unit 2

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 free-response question

Probability rules, conditional probability and independence, discrete random variables, the binomial distribution, and sampling distributions. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Free-response practice
  5. Common mistakes
  6. Keep going

Key equations for probability, random variables and probability distributions

P(A or B) = P(A) + P(B) − P(A and B)addition rule
P(A | B) = P(A and B) / P(B)conditional probability
E(X) = Σ x P(x)expected value
P(X = k) = nCk pk(1 − p)n−kbinomial probability
σx̄ = σ / √nSD of the sample mean

The formulas and tables are provided in the exam. Answers within 2.5% are marked correct (5% for probabilities), so values from a table or a calculator both count. The full sheet is in our AP Statistics formula sheet.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-up
P(A) = 0.5, P(B) = 0.4 and P(A and B) = 0.2. Find P(A or B).

Hint

Use the addition rule, so the overlap is not counted twice.

Worked solution
  1. P(A or B) = 0.5 + 0.4 − 0.2.
  2. = 0.7.
Q2Warm-up
A random variable X takes the values 0, 1 and 2 with probabilities 0.2, 0.5 and 0.3. Find E(X).

Hint

Multiply each value by its probability and add.

Worked solution
  1. E(X) = 0(0.2) + 1(0.5) + 2(0.3).
  2. E(X) = 1.1.
Q3Warm-up
Events A and B are independent if:

Hint

Independence means knowing B happened does not change the chance of A.

Worked solution
  1. If P(A | B) = P(A), learning that B happened tells you nothing about A.
  2. That is independence: B. Option D describes mutually exclusive events.

Exam-style questions

Questions at the level of the AP Statistics exam. The ones marked Challenge combine more than one idea.

Q4Exam-style
In a school, 30% of students play a sport, and 12% play a sport and play an instrument. What is the probability that a student who plays a sport also plays an instrument?

Hint

This is a conditional probability: given that they play a sport.

Worked solution
  1. P(instrument | sport) = P(both)/P(sport).
  2. = 0.12/0.30 = 0.4.
Q5Exam-style
A basketball player makes 30% of her three-point attempts, independently. In 10 attempts, what is the probability she makes exactly 3?

Hint

Binomial with n = 10 and p = 0.3.

Worked solution
  1. P(X = 3) = 10C3(0.3)3(0.7)7 = 120 × 0.027 × 0.0824.
  2. P(X = 3) = 0.267.
Q6Exam-style
Each of 5 independent light bulbs is faulty with probability 0.2. What is the probability that at least one is faulty?

Hint

Use the complement: 1 − P(none faulty).

Worked solution
  1. P(none faulty) = 0.85 = 0.328.
  2. P(at least one) = 1 − 0.328 = 0.672.
Q7Exam-style
X is binomial with n = 100 and p = 0.4. Find the standard deviation of X.

Hint

σ = √(np(1 − p)).

Worked solution
  1. σ = √(100 × 0.4 × 0.6) = √24.
  2. σ = 4.90.
Q8Challenge
A population has mean 50 and standard deviation 12. For random samples of size 36, find the probability that the sample mean is greater than 53.

Hint

Find the standard deviation of the sample mean first. With n = 36 the central limit theorem lets you use a normal model.

Worked solution
  1. σx̄ = 12/√36 = 2.
  2. z = (53 − 50)/2 = 1.5.
  3. P(x̄ > 53) = P(Z > 1.5) = 0.0668.

Free-response practice

One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.

Probability and random variables style · 10 points · about 22 minutes

At a school fair, a game costs $2 to play. A player spins a fair spinner with 10 equal sections: 1 section wins $10, 3 sections win $2, and the rest win nothing. Let W be the player's net gain from one game.

  1. (a) Write down the probability distribution of W.
  2. (b) Find E(W) and interpret it in context.
  3. (c) A player plays 8 games. Find the probability that they win the $10 prize at least once.
  4. (d) Is the number of games until the first $10 win a binomial random variable? Explain.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

W = 8 with probability 0.1; W = 0 with probability 0.3; W = −2 with probability 0.6.

Part (b)

E(W) = 8(0.1) + 0(0.3) − 2(0.6) = −0.4. Over many games, a player loses an average of 40 cents a game.

Part (c)

Binomial, n = 8, p = 0.1. P(at least one) = 1 − 0.98 = 0.570.

Part (d)

No. A binomial variable has a fixed number of trials; here the number of trials is not fixed.

Your mark: 0 of 10

Common mistakes in probability, random variables and probability distributions

Keep going

Revise the ideas behind these questions:

Then try real exam questions: AP Statistics past papers, with official scoring guidelines. · All AP Statistics units