• 53 questions

AP Statistics Random Variables Practice Test

Unofficial practice on random variables in AP Statistics: probability distributions, expected value, standard deviation and the binomial distribution.

What is a random variable?

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Unofficial practice questions, not affiliated with the College Board. Bank checked October 11, 2026.

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All 53 questions

Topic 2.8: Random variables and probability distributions 13

  • A cumulative probability table for a discrete random variable X gives P(X ≤ 2) = 0.55 and P(X ≤ 3) = 0.82. X takes whole-number values. What is P(X = 3)? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.5
  • A discrete random variable X takes the values 0, 1, 2 and 3 with probabilities 0.10, 0.35, 0.30 and k. What is k? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.2
  • A fair coin is tossed twice and X is the number of heads. What is P(X = 1)? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.3
  • A fair six-sided die is rolled once and X is the number that comes up. What is P(X > 4)? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.3
  • A student simulates rolling two dice 500 times. The sum is 7 in 90 of the 500 trials. What is the simulation's estimate of P(sum = 7)? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.3
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Topic 2.8: Random variables and probability distributions, continued

  • In which forms can the probability distribution of a discrete random variable be represented? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.4
  • What is a random variable? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.1
  • What is the cumulative probability at the largest possible value of a discrete random variable? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.5
  • Which list of probabilities for X = 0, 1 and 2 cannot be a probability distribution? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.2
  • Which of these is a discrete random variable? College Board, AP Statistics Course and Exam Description, topic 2.9, essential knowledge 2.9.A.2
  • X has P(X = 0) = 0.15, P(X = 1) = 0.25, P(X = 2) = 0.40 and P(X = 3) = 0.20. What is P(X < 2)? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.2
  • X has P(X = 0) = 0.15, P(X = 1) = 0.25, P(X = 2) = 0.40 and P(X = 3) = 0.20. What is P(X ≥ 2)? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.2
  • X has P(X = 0) = 0.2, P(X = 1) = 0.5 and P(X = 2) = 0.3. What does the cumulative probability distribution show at the value 1? College Board, AP Statistics Course and Exam Description, topic 2.8, essential knowledge 2.8.A.5

Topic 2.9: Mean and standard deviation of a random variable 13

  • A charity sells 500 tickets at $4 each, and one ticket wins $1,000. What is the expected net gain from buying one ticket? Wikipedia, "Expected value"; course topic 2.9
  • A machine jams 0, 1 or 2 times in a shift, with probabilities 0.5, 0.3 and 0.2. What is the expected number of jams in a shift? College Board, AP Statistics Course and Exam Description, topic 2.9, essential knowledge 2.9.A.2
  • A scratch card gives a prize of $10 with probability 0.1, a prize of $2 with probability 0.3 and no prize the rest of the time. What is the mean prize of one card? Wikipedia, "Expected value"; course topic 2.9
  • The mean of the probability distribution of a random variable is which kind of number? College Board, AP Statistics Course and Exam Description, topic 2.9, essential knowledge 2.9.A.1
  • Random variables A and B have the same mean. The likely values of A sit close to the mean; the likely values of B are spread far to both sides. Which has the larger standard deviation? College Board, AP Statistics Course and Exam Description, topic 2.9, essential knowledge 2.9.A.3
  • The variance of a random variable is 9. What is its standard deviation? Wikipedia, "Variance"; course topic 2.9
  • What is the expected value of the number that comes up on one roll of a fair six-sided die? College Board, AP Statistics Course and Exam Description, topic 2.9, essential knowledge 2.9.A.2
  • X is the number of calls a help desk gets in an hour, and the expected value of X is 4.2. Which interpretation is correct? College Board, AP Statistics Course and Exam Description, topic 2.9, essential knowledge 2.9.A.2
  • X is the number of customers a shop gets in a day, with mean 120 and standard deviation 15. Which interpretation of the standard deviation is correct? College Board, AP Statistics Course and Exam Description, topic 2.9, essential knowledge 2.9.A.3
  • X is the number of pets in a household, with P(0) = 0.3, P(1) = 0.4, P(2) = 0.2 and P(3) = 0.1. What is the mean of X? College Board, AP Statistics Course and Exam Description, topic 2.9, essential knowledge 2.9.A.2
  • X takes the value 0 with probability 0.8 and the value 10 with probability 0.2. What is the standard deviation of X? Wikipedia, "Variance"; course topic 2.9
  • X takes the values 0 and 2, each with probability 0.5. What is the standard deviation of X? College Board, AP Statistics Course and Exam Description, topic 2.9, essential knowledge 2.9.A.3
  • X takes the values 1, 2 and 3 with probabilities 0.25, 0.50 and 0.25. To two decimal places, what is the standard deviation of X? College Board, AP Statistics Course and Exam Description, topic 2.9, essential knowledge 2.9.A.3

Topic 2.10: The binomial distribution 17

  • A player makes 70% of her free throws, and each shot is independent. X is the number she makes in 20 shots, so the mean of X is 14. Which interpretation is correct? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.C.1
  • A quiz has 10 multiple-choice questions with 4 choices each, one of them right. A student guesses at random on every question, and X is the number right. What are n and p? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.A.1
  • A simulation runs 1,000 sets of 10 free throws for a 70% shooter. In 380 of the sets she makes 8 or more. What is the simulation's estimate of P(X ≥ 8)? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.D.1
  • A spinner lands on red half the time. In 4 spins, what is the probability that it lands on red exactly twice? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.E.1
  • Each of 4 seeds sprouts with probability 0.5, independently of the others. What is the probability that at most 1 of them sprouts? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.E.1
  • Five cards are dealt from a deck without replacement and X is the number of red cards. Why is X not binomial? Wikipedia, "Binomial distribution"; course topic 2.10
  • For a binomial random variable with a fixed number of trials n, which probability of success gives the largest standard deviation? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.B.1
  • In the binomial probability function, the probability of exactly x successes in n trials starts with the count "n choose x". What does that count give? Wikipedia, "Binomial distribution"; course topic 2.10
  • Which of these is a binomial random variable? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.A.1
  • X is binomial with n = 3 and p = 0.2. What is P(X = 0)? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.E.1
  • X is binomial with n = 5 and p = 0.1. To three decimal places, what is P(X ≥ 1)? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.E.1
  • X is binomial with n = 5 and p = 0.2. To three decimal places, what is P(X = 2)? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.E.1
  • X is binomial with n = 10 and p = 0.25. To two decimal places, what is the standard deviation of X? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.B.1
  • X is binomial with n = 48 and p = 0.25. What is the standard deviation of X? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.B.1
  • X is binomial with n = 50 and p = 0.2. What is the mean of X? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.B.1
  • X is binomial with n = 100 and p = 0.5. What is the standard deviation of X? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.B.1
  • X is binomial with n = 400 and p = 0.5, so its mean is 200 and its standard deviation is 10. How many standard deviations above the mean is a result of 230? College Board, AP Statistics Course and Exam Description, topic 2.10, essential knowledge 2.10.B.1

Topic 2.11: The normal distribution 10

  • Two normal curves have the same mean. One has standard deviation 2 and the other has standard deviation 5. Which curve is taller and more concentrated around the mean? College Board, AP Statistics Course and Exam Description, topic 2.11, essential knowledge 2.11.A.5
  • What are the mean and the standard deviation of the standard normal distribution? College Board, AP Statistics Course and Exam Description, topic 2.11, essential knowledge 2.11.B.1
  • What is the total area under a normal curve? College Board, AP Statistics Course and Exam Description, topic 2.11, essential knowledge 2.11.D.1
  • Which values can a continuous random variable take? College Board, AP Statistics Course and Exam Description, topic 2.11, essential knowledge 2.11.A.1
  • X is a random variable with an approximately normal distribution. How is the probability that X falls between two values found? College Board, AP Statistics Course and Exam Description, topic 2.11, essential knowledge 2.11.D.1
  • X is normal with mean 70 and standard deviation 4. By the empirical rule, about what percent of values lie between 70 and 78? College Board, AP Statistics Course and Exam Description, topic 2.11, essential knowledge 2.11.C.1
  • X is normal with mean 70 and standard deviation 4. What is the z-score of the value 78? Wikipedia, "Standard score"; course topic 2.11
  • X is normal with mean 100 and standard deviation 15. By the empirical rule, about what percent of values are greater than 130? College Board, AP Statistics Course and Exam Description, topic 2.11, essential knowledge 2.11.C.1
  • X is normal with mean 100 and standard deviation 15. By the empirical rule, about what percent of values are less than 85? College Board, AP Statistics Course and Exam Description, topic 2.11, essential knowledge 2.11.C.1
  • X is normal with mean 100 and standard deviation 15. By the empirical rule, about what percent of values lie between 85 and 115? College Board, AP Statistics Course and Exam Description, topic 2.11, essential knowledge 2.11.C.1

How this list was compiled: Our own questions, written to the random variable topics of Unit 2 (Probability, Random Variables, and Probability Distributions) of the College Board's AP Statistics Course and Exam Description, effective fall 2026, https://apcentral.collegeboard.org/media/pdf/ap-statistics-course-and-exam-description.pdf, read 2026-10-11: 2.8 Introduction to Random Variables and Probability Distributions, 2.9 Parameters of Random Variables, 2.10 The Binomial Distribution and 2.11 The Normal Distribution. Each rule was read on 2026-10-11 in the essential knowledge statement or the Wikipedia article named under its question, and every worked answer was recomputed from the numbers in its question. Licence: Own work. Short quotations from the public documents named under each question.

How it works

This AP Statistics random variables practice test has 53 questions of our own. It is unofficial and not affiliated with the College Board, which owns AP.

Read the full guide

Where random variables sit now

The College Board revised AP Statistics for the 2026-27 school year. Its course description, effective fall 2026 and read October 11, 2026, puts random variables in Unit 2. The bank follows four topics: 2.8, random variables and their distributions (13 questions); 2.9, mean and standard deviation (13); 2.10, the binomial distribution (17); and 2.11, the normal distribution (10). The description gives topics no weights, so the spread is ours.

How the answers are checked

Under each answer is the essential knowledge statement or the Wikipedia article its rule was read in. Every worked answer was computed a second time from the numbers in its question.

  • For an expected value, multiply each value by its probability and add. The result need not be a value the variable can take.
  • A binomial count needs two outcomes, a set number of trials, independent trials and one probability of success.
  • For "at least one", find the probability of none and subtract it from 1.

Questions and answers

What unit is random variables in AP Statistics?

Unit 2, Probability, Random Variables, and Probability Distributions, in the course description effective fall 2026. The unit carries 15 to 25% of the multiple-choice section and about 24 class periods.

Does this AP Statistics random variables practice test cover binomial and geometric distributions?

Binomial, yes: topic 2.10 has 17 questions. Geometric, no. The course description effective fall 2026 has no topic on the geometric distribution. One binomial question does show why counting trials until the first success is not binomial.

What makes a random variable binomial?

It counts the successes in a set number of repeated, independent trials that each have two possible outcomes and the same probability of success. Its mean is np and its standard deviation is the square root of np(1 − p).

What is the difference between a discrete and a continuous random variable?

A discrete random variable takes values that are countable or finite, such as a number of defects. A continuous one can take any value within a domain, such as a time, and its probabilities belong to intervals.

Are these real AP Statistics exam questions?

No. None of the College Board's questions appears on this page, in its own words or in others. Each question was written for the page from the topics that the course description lists.

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