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AP Statistics Practice Quiz: Introduction to Random Variables and Probability Distributions

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 16 questions to check your progress.

Question 1 of 16

Which of the following best defines a random variable?

All Questions (16)

Which of the following best defines a random variable?

A) A variable that is always a whole number.

B) The average outcome of a random event.

C) A variable whose values are the numerical outcomes of random behavior.

D) A fixed value that does not change.

Correct Answer: C

Based on the provided content, the values of a random variable are the numerical outcomes of random behavior. The other options are incorrect definitions.

A key characteristic of a discrete random variable is that it can take on:

A) any value within a given range.

B) only positive values.

C) a countable number of values.

D) an infinite, uncountable number of values.

Correct Answer: C

The content states that a discrete random variable can take a countable number of values. An uncountable number of values would describe a continuous random variable, which is not covered in the provided text.

For any discrete random variable, what must be true about the sum of the probabilities of all its possible values?

A) The sum must be equal to 0.

B) The sum must be equal to 1.

C) The sum must be less than 1.

D) The sum depends on the number of outcomes.

Correct Answer: B

A fundamental rule for any probability distribution, as stated in the content, is that the sum of the probabilities for all possible outcomes of a discrete random variable must be equal to 1.

Which of the following is NOT a valid way to represent a probability distribution?

A) A table

B) A graph

C) A function

D) A single numerical value

Correct Answer: D

The content explicitly mentions that a probability distribution can be represented by a graph, table, or function. A single numerical value, like a mean or a standard deviation, can describe a feature of the distribution but cannot represent the entire distribution itself.

A cumulative probability distribution for a random variable X shows the probability that:

A) X is exactly equal to a specific value.

B) X is greater than a specific value.

C) X is less than or equal to each value.

D) X is the most likely outcome.

Correct Answer: C

The definition provided in the content states that a cumulative probability distribution shows the probability of being less than or equal to each value of the random variable.

When we interpret a probability distribution, what information can we gather about the population?

A) The exact size of the population.

B) The specific individuals in the population.

C) The shape, center, and spread of the population's values.

D) The cause of the random behavior in the population.

Correct Answer: C

The content specifies that interpreting a probability distribution provides information about the shape, center, and spread of a population.

Let X be a discrete random variable with the following partial probability distribution: P(X=1)=0.2, P(X=2)=0.4, P(X=3)=0.1. If 1, 2, 3, and 4 are the only possible values of X, what is P(X=4)?

A) 0.1

B) 0.2

C) 0.3

D) 0.4

Correct Answer: C

The sum of all probabilities in a discrete probability distribution must equal 1. The sum of the given probabilities is 0.2 + 0.4 + 0.1 = 0.7. Therefore, P(X=4) must be 1 - 0.7 = 0.3.

Which of the following tables represents a valid probability distribution for a discrete random variable X?

A) X: 0, 1, 2; P(X): 0.3, 0.4, 0.4

B) X: 1, 2, 3; P(X): 0.5, -0.1, 0.6

C) X: 0, 1, 2; P(X): 0.5, 0.2, 0.3

D) X: 1, 2, 3; P(X): 0.2, 0.3, 0.4

Correct Answer: C

A valid probability distribution requires two conditions: all probabilities must be between 0 and 1, and the sum of all probabilities must equal 1. Option C is the only one that meets both criteria (0.5 + 0.2 + 0.3 = 1.0). Option A sums to 1.1. Option B includes a negative probability. Option D sums to 0.9.

The primary purpose of a probability distribution is to:

A) prove that an event is truly random.

B) list every possible numerical outcome of a random behavior and its associated probability.

C) calculate the average of a dataset.

D) determine the sample size needed for an experiment.

Correct Answer: B

A probability distribution for a discrete random variable links each possible numerical outcome with its probability of occurrence. This aligns with the content about representing the distribution and defining a random variable.

Consider the probability distribution where X can be 0, 1, or 2, with P(X=0)=0.5, P(X=1)=0.3, and P(X=2)=0.2. What is the cumulative probability P(X ≤ 1)?

A) 0.3

B) 0.5

C) 0.8

D) 1.0

Correct Answer: C

The cumulative probability P(X ≤ 1) is the probability that X is less than or equal to 1. This is calculated by summing the probabilities of all outcomes that meet this condition: P(X=0) + P(X=1) = 0.5 + 0.3 = 0.8.

A fair six-sided die is rolled. Let the random variable Y be the number showing on the top face. The values that Y can take are:

A) countable and finite.

B) uncountable.

C) only even numbers.

D) infinite.

Correct Answer: A

The outcomes are the numbers {1, 2, 3, 4, 5, 6}. This is a countable and finite set of values, which is characteristic of a discrete random variable.

A probability distribution is described as being skewed to the right. This description of the distribution's shape is part of:

A) calculating a cumulative probability.

B) verifying the sum of probabilities is 1.

C) defining the random variable.

D) interpreting the probability distribution.

Correct Answer: D

According to the content, interpreting a probability distribution provides information about its shape, center, and spread. Describing the shape as 'skewed to the right' is a key part of this interpretation.

A table that lists P(X ≤ k) for each possible value k of a random variable X is called a:

A) frequency distribution.

B) probability distribution.

C) cumulative probability distribution.

D) variable distribution.

Correct Answer: C

The content defines a cumulative probability distribution as one that shows the probability of the random variable being less than or equal to each value, which is precisely what P(X ≤ k) represents.

If a probability distribution is represented by a function, what does the function provide?

A) The average value of the random variable.

B) The probability for each possible value of the random variable.

C) The total number of outcomes.

D) The range of the random variable.

Correct Answer: B

The content states that a probability distribution can be represented by a function. The purpose of this function (or a table or graph) is to specify the probability for each of the random variable's possible values.

The number of defective items in a batch of 10 is a discrete random variable. Why?

A) Because the number of defective items can be any value between 0 and 10.

B) Because the outcomes are numerical and result from random behavior.

C) Because the possible values (0, 1, 2, ..., 10) are countable.

D) Because the probability of finding a defective item is always small.

Correct Answer: C

While the outcomes are numerical (B), the specific reason it is a *discrete* random variable is that it can only take on a countable number of values (the integers from 0 to 10).

Let X be a random variable representing the number of heads in two coin flips. P(X=0)=0.25, P(X=1)=0.50, P(X=2)=0.25. A student calculates the center of this distribution to be 1. What does this value represent?

A) The most frequent outcome in a single trial.

B) The long-run average outcome if the experiment were repeated many times.

C) The cumulative probability of the distribution.

D) The spread of the distribution.

Correct Answer: B

The center of a probability distribution (in this case, the mean or expected value) represents the theoretical long-run average of the random variable's outcomes. This is a key aspect of interpreting the center of a population distribution.