AP Statistics Probability Practice with Answers

AP Statistics probability practice with worked answers covering probability rules, conditional probability, random variables, and binomial settings.

Probability is one of the most important bridges in AP Statistics. It connects exploratory data analysis to sampling distributions and inference. If a student can use probability rules, conditional probability, random variables, and binomial models accurately, the later inference units become much easier.

College Board lists AP Statistics as an introductory college-level statistics course, and the current AP Statistics course page identifies Unit 4, Probability, Random Variables, and Probability Distributions, as part of the course framework. The AP Statistics exam includes multiple-choice and free-response sections, and the free-response section includes a probability and sampling distributions focus. Always check the official AP Statistics course page and AP Statistics exam page for current exam details.

ViewMath is an independent publisher and is not affiliated with or endorsed by College Board or the AP Program.

Probability Skills to Master

  • Basic rules: complements, union, intersection, and mutually exclusive events.
  • Conditional probability: probability of A given B and interpreting two-way tables.
  • Independence: knowing when P(A and B) = P(A)P(B) applies.
  • Random variables: expected value, variance, and standard deviation.
  • Binomial settings: fixed trials, independent trials, two outcomes, and constant probability.

AP Statistics Probability Practice

Basic Probability Rules

  1. A card is drawn from a standard 52-card deck. What is the probability that the card is a heart?
  2. In a class of 30 students, 18 play a sport, 12 play an instrument, and 7 do both. What is the probability that a randomly selected student plays a sport or an instrument?
  3. Events A and B are independent. P(A) = 0.35 and P(B) = 0.20. What is P(A and B)?

Conditional Probability

  1. A bag contains 5 red marbles and 4 blue marbles. Two marbles are selected without replacement. Given that the first marble was red, what is the probability that the second marble is red?
  2. A survey of 200 students found that 120 take math, 90 take science, and 50 take both. Given that a student takes science, what is the probability the student also takes math?

Random Variables

  1. A game pays $6 if you roll a 6 on a fair die and pays $0 otherwise. It costs $1 to play. Let X be the net gain. Find E(X).
  2. A random variable X has values 0, 1, and 2 with probabilities 0.2, 0.5, and 0.3. Find E(X).

Binomial Probability

  1. A fair coin is flipped 10 times. What is the probability of getting exactly 6 heads?
  2. A multiple-choice question has 4 choices. A student guesses on 8 independent questions. Let X be the number correct. Find the mean and standard deviation of X.
  3. A basketball player makes 70% of free throws. What is the probability the player makes exactly 4 of the next 5 free throws?

Mixed AP-Style Prompts

  1. A medical test is positive for 96% of people with a condition and for 5% of people without the condition. In a population of 10,000 people where 2% have the condition, estimate the probability that a person with a positive test actually has the condition.
  2. A spinner lands on red with probability 0.30. The spinner is spun until it lands on red for the first time. Is this a binomial setting? Explain.
  3. A random variable has mean 12 and standard deviation 3. Another independent random variable has mean 8 and standard deviation 4. What are the mean and standard deviation of their sum?
  4. In a two-way table, 40 of 100 students are seniors, 25 students have parking permits, and 18 seniors have parking permits. Are being a senior and having a parking permit independent? Use probabilities to justify your answer.

Answer Key with Explanations

  1. There are 13 hearts out of 52 cards, so P(heart) = 13/52 = 1/4.
  2. P(sport or instrument) = (18 + 12 – 7) / 30 = 23/30.
  3. Because A and B are independent, P(A and B) = 0.35 x 0.20 = 0.07.
  4. After one red marble is removed, 4 red and 4 blue remain. P(red second | red first) = 4/8 = 1/2.
  5. P(math | science) = students taking both / students taking science = 50/90 = 5/9.
  6. Net gain is $5 if you roll a 6 and -$1 otherwise. E(X) = (1/6)(5) + (5/6)(-1) = 0. This is a fair game.
  7. E(X) = 0(0.2) + 1(0.5) + 2(0.3) = 1.1.
  8. P(X = 6) = C(10,6)(0.5)^6(0.5)^4 = C(10,6)(0.5)^10 = 210/1024, about 0.205.
  9. This is binomial with n = 8 and p = 0.25. Mean = np = 2. Standard deviation = sqrt(np(1-p)) = sqrt(8 x 0.25 x 0.75) = sqrt(1.5), about 1.225.
  10. P(X = 4) = C(5,4)(0.7)^4(0.3)^1 = 5 x 0.2401 x 0.3 = 0.36015.
  11. About 200 people have the condition, and 96% of them test positive: 192 true positives. About 9,800 people do not have the condition, and 5% of them test positive: 490 false positives. Positive tests total 682, so the probability is 192 / 682, about 0.281.
  12. No. It has two outcomes and a constant probability, but the number of trials is not fixed in advance. This is a geometric setting, not binomial.
  13. The mean of the sum is 12 + 8 = 20. For independent random variables, variances add: 3^2 + 4^2 = 9 + 16 = 25, so the standard deviation is 5.
  14. P(permit) = 25/100 = 0.25. P(permit | senior) = 18/40 = 0.45. Since these are not equal, the events are not independent.

Common AP Statistics Probability Mistakes

  • Adding probabilities when events overlap. Use P(A or B) = P(A) + P(B) – P(A and B).
  • Assuming independence. Sampling without replacement usually changes probabilities unless the sample is very small relative to the population.
  • Confusing conditional direction. P(A given B) is not the same as P(B given A).
  • Using binomial formulas too quickly. Check the binomial conditions before calculating.
  • Forgetting context. AP Statistics scoring rewards interpretation, not just numeric answers.

Diagnostic Checks Before You Move On

A student is ready to leave probability review when they can do three things without prompting: name the probability model, define the event in words, and write one sentence interpreting the result. For example, a binomial answer should not stop at 0.36015. A stronger response says that, under the model, there is about a 36.0% chance the player makes exactly 4 of the next 5 free throws.

If the student can calculate but cannot choose a model, mix together conditional probability, random variable, binomial, and geometric prompts. If the student chooses models correctly but makes arithmetic errors, slow down and require formula substitution before calculator work.

One-Week Probability Review Plan

Day Focus Practice
1 Complements and unions 20 short probability-rule questions.
2 Two-way tables Conditional probability from tables.
3 Independence Compare P(A and B) with P(A)P(B).
4 Random variables Expected value and interpretation.
5 Binomial models Identify conditions and calculate probabilities.
6 Mixed AP-style questions Short response practice with written explanations.
7 Error review Redo missed problems without notes.

Strong probability work is slow at first because students must name the model before calculating. That habit pays off later when inference questions depend on choosing the correct probability structure.

Using ViewMath for AP Statistics Probability

Use this practice set as a diagnostic, then assign focused ViewMath statistics work by topic: probability rules first, conditional probability second, random variables third, and binomial settings last. After a focused review block, return to mixed AP-style questions so students practice deciding which tool fits the context.

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