Probability, random variables and distributions practice questions (AP Statistics)

Sample questions

All 6 free AP Statistics questions on this unit, from the free AP Statistics Practice Exam and the free AP Statistics Free-Response Practice Questions (FRQ) with Answers. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 12 marks · Conditional probability
   At a school of 200 students, 120 are juniors and 80 are seniors. Forty-eight juniors and 40 seniors have part-time jobs. If a randomly selected student is a senior, the probability that they have a part-time job is:
   • A.
     $0.20$
   • B.
     $0.44$
   • C.
     $0.50$
   • D.
     $0.45$
   • E.
     $0.40$
   Show answerFrom the free AP Statistics Practice Exam →
2. Question 22 marks · Independence
   Events $A$ and $B$ have $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cap B) = 0.2$. Which is true?
   • A.
     $A$ and $B$ are mutually exclusive
   • B.
     A$ and $B$ are independent, since $P(A)P(B) = P(A \cap B)

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 2: A$̲ and $B$ are in…
   • C.
     A$ and $B$ are dependent, since $P(A \cap B) \neq 0

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 2: A$̲ and $B$ are de…
   • D.
     $P(A \cup B) = 0.9$
   • E.
     $A$ and $B$ cannot both occur
   Show answerFrom the free AP Statistics Practice Exam →
3. Question 32 marks · Binomial setting
   Which of the following scenarios describes a binomial random variable?
   • A.
     Counting made baskets in 20 independent free throws, each with success probability 0.7
   • B.
     Counting draws from a deck WITHOUT replacement until the first ace appears
   • C.
     Recording the time until a light bulb fails
   • D.
     Counting cars passing an intersection in an hour
   • E.
     Measuring the average height of 30 random students
   Show answerFrom the free AP Statistics Practice Exam →
4. Question 42 marks · Expected value
   A raffle sells 500 tickets at 2 dollars each, with a single prize worth 400 dollars. The expected net gain from buying one ticket is:
   • A.
     a loss of 1.20 dollars
   • B.
     a gain of 0.80 dollars
   • C.
     a loss of 2.00 dollars
   • D.
     a loss of 0.80 dollars
   • E.
     exactly zero
   Show answerFrom the free AP Statistics Practice Exam →
5. Question 58 marks · Discrete random variables
   Expected value. An insurance company sells a one-year policy for 150 dollars. Based on historical data, in a given year a policyholder files no claim with probability 0.95, a minor claim paying out 1,000 dollars with probability 0.04, and a major claim paying out 10,000 dollars with probability 0.01.
   (a)

   Let $X$ be the payout on a randomly selected policy. Construct the probability distribution of $X$ and verify it is valid.

   [2]
   (b)

   Compute the expected payout $E(X)$ and interpret it in context.

   [2]
   (c)

   Compute the standard deviation of $X$.

   [2]
   (d)

   What is the company's expected profit per policy, and why can the company rely on this number even though individual policies vary so much?

   [2]
   Show answerFrom the free AP Statistics Practice Exam →
6. Question 64 marks · Probability & random variables
   Probability. An online retailer ships every order from one of two warehouses. Seventy percent of orders ship from warehouse A and thirty percent from warehouse B. Records show that 5 percent of the orders shipped from warehouse A arrive late, and 12 percent of the orders shipped from warehouse B arrive late. Assume orders are independent of one another.
   (a)

   Find the probability that a randomly selected order arrives late.

   [1]
   (b)

   Given that an order arrives late, find the probability that it was shipped from warehouse B.

   [1]
   (c)

   Are the events "the order arrives late" and "the order was shipped from warehouse B" independent? Justify your answer using probabilities.

   [1]
   (d)

   Let $X$ be the number of late orders among 10 randomly selected orders. Find the probability that at least one of the 10 orders arrives late, and find the mean and standard deviation of $X$.

   [1]
   Show answerFrom the free AP Statistics Free-Response Practice Questions (FRQ) with Answers →

Key terms

The definitions and formulas AP Statistics expects for probability, random variables and distributions, typeset as they appear on the paper.

Addition rule

The probability that at least one of two events happens. For mutually exclusive events the intersection term is zero.

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

Conditional probability

The probability of $A$ given that $B$ occurred, found by restricting attention to the cases where $B$ happened. Two-way tables and tree diagrams organise the calculation.

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$

Independence

Events are independent when knowing one does not change the probability of the other. Check it with the definition using numbers from the table; do not assert it.

$$P(A \mid B) = P(A) \iff P(A \cap B) = P(A)\,P(B)$$

Mean and standard deviation of a discrete random variable

The expected value is the probability-weighted average of the outcomes; the variance is the probability-weighted average squared deviation.

$$\mu_X = \sum x_i p_i, \qquad \sigma_X^2 = \sum (x_i - \mu_X)^2 p_i$$

Combining random variables

Means add (or subtract); variances add for independent variables even when the variables are subtracted. Standard deviations never add directly.

$$\mu_{X \pm Y} = \mu_X \pm \mu_Y, \qquad \sigma^2_{X \pm Y} = \sigma_X^2 + \sigma_Y^2 \;\text{(independent)}$$

Binomial distribution

The count of successes in a fixed number $n$ of independent trials, each with the same success probability $p$ (check the four conditions: binary, independent, fixed number, same probability).

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \qquad \mu = np, \qquad \sigma = \sqrt{np(1-p)}$$

Geometric distribution

The number of trials up to and including the first success, with no fixed number of trials.

$$P(X = k) = (1-p)^{k-1} p, \qquad \mu = \frac{1}{p}$$

Frequently asked questions

Other AP Statistics topics

• Exploring one-variable data
• Exploring two-variable data
• Collecting data
• Sampling distributions
• Inference for proportions
• Inference for means
• Chi-square tests
• Inference for slopes

All AP Statistics topics · AP Statistics exam guide · All free practice exams