AP Statistics Quiz: Random Variables And Probability Distributions

What this quiz covers

This quiz focuses on Random Variables And Probability Distributions, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. 0.4
2. 0.5
3. 0.7 (correct answer)
4. 0.9

Explanation: The cumulative probability $P(X \le 2)$ is the sum of the probabilities for all outcomes less than or equal to 2. This is calculated as $P(X=0) + P(X=1) + P(X=2) = 0.1 + 0.2 + 0.4 = 0.7$.

Question 2

1. 0.04
2. 0.06
3. 0.07 (correct answer)
4. 0.93

Explanation: The probability that a watch has at least two defects is $P(X \ge 2)$. This is calculated by summing the probabilities for X=2, X=3, and X=4. So, $P(X \ge 2) = P(X=2) + P(X=3) + P(X=4) = 0.04 + 0.02 + 0.01 = 0.07$.

Question 3

1. 0.55 (correct answer)
2. 0.70
3. 0.85
4. 0.95

Explanation: We need to find the probability $P(1 < Y \le 4)$. This corresponds to the outcomes Y=2, Y=3, and Y=4. We sum their probabilities: $P(Y=2) + P(Y=3) + P(Y=4) = 0.30 + 0.15 + 0.10 = 0.55$.

Question 4

1. 1/10 (correct answer)
2. 1/4
3. 1
4. 10

Explanation: For a valid probability distribution, the sum of all probabilities must be 1. We sum the probabilities for each value of y: $P(Y=1)+P(Y=2)+P(Y=3)+P(Y=4) = c(1) + c(2) + c(3) + c(4) = 1$. This simplifies to $10c = 1$, so the constant $c$ must be $1/10$.

Question 5

1. $B$ is the probability that a visitor checks out at least 1 book.
2. $B$ is the number of books checked out by a randomly selected visitor in one visit. (correct answer)
3. $B$ is the event that a visitor checks out exactly 0 books.
4. $B$ is the total number of books in the library.
5. $B$ is the list of outcomes (0 through 4) rather than a random variable.

Explanation: This question tests comprehension of random variables in a library usage context. The random variable B is defined as "the number of books checked out by a randomly selected visitor in one visit," with possible values 0, 1, 2, 3, or 4. Choice A incorrectly interprets B as a probability of checking out books rather than the count itself. Choice C mistakes B for a specific event (checking out 0 books) instead of recognizing it as a variable with multiple outcomes. Choice D absurdly confuses the visitor's checkout count with the library's total collection. Choice E suggests B is just a list of numbers rather than a proper random variable with associated probabilities. A discrete random variable maps random outcomes to numerical values—here, counting books borrowed by one library visitor.

Question 6

1. $P$ is the event that a household owns exactly 2 pets.
2. $P$ is the probability that a household owns pets.
3. $P$ is the number of pets owned by a randomly selected household. (correct answer)
4. $P$ is the total number of pets in the entire sample of households.
5. $P$ is the list of possible pet counts without considering their probabilities.

Explanation: This question evaluates comprehension of random variables in a household survey context. The random variable P represents "the number of pets owned by a randomly selected household," with possible values 0, 1, 2, 3, or 4. Choice A incorrectly treats P as a specific event (owning exactly 2 pets) rather than a variable with multiple values. Choice B misinterprets P as a probability of pet ownership rather than the count of pets. Choice D confuses the individual household's pet count with the total across all sampled households. Choice E describes P as merely a list without recognizing its role as a random variable with associated probabilities. A discrete random variable assigns numerical values to random outcomes—here, the number of pets in one randomly selected household.

Question 7

1. $X$ is the probability that a customer buys pastries.
2. $X$ is the number of pastries purchased by a randomly selected customer during a visit. (correct answer)
3. $X$ is the list of possible outcomes for pastry purchases: $0,1,2,3$.
4. $X$ is the event that a customer buys at least 1 pastry.
5. $X$ is the probability that a customer buys exactly 2 pastries.

Explanation: This question tests understanding of what a random variable represents in probability distributions. A random variable is a numerical outcome of a random phenomenon, not a probability or event. In this context, X represents the actual number of pastries purchased by a randomly selected customer (choice B), which can take values 0, 1, 2, or 3. Choice A incorrectly describes X as a probability, choice C describes X as a list rather than a variable, choice D describes X as an event, and choice E again confuses X with a probability value. Remember that a random variable assigns numerical values to outcomes of random experiments.

Question 8

1. $X$ is the number of damaged items in a randomly selected box of 5 items. (correct answer)
2. $X$ is the probability that an item is damaged.
3. $X$ is the event that the box has no damaged items.
4. $X$ is the probability that the box has exactly 2 damaged items.
5. $X$ is the list of probabilities in the distribution table.

Explanation: This question evaluates understanding of random variables in quality control contexts. The random variable X represents the count of damaged items found in a randomly selected box containing 5 items (choice A). X can take values 0, 1, 2, 3, 4, or 5, representing the possible number of damaged items. Choice B incorrectly describes X as a probability rather than a count, choice C describes X as an event (a yes/no outcome), choice D again confuses X with a specific probability value, and choice E misidentifies X as the list of probabilities. Remember that random variables represent measurable numerical outcomes, not probabilities or events themselves.

Question 9

1. $X$ is the probability that exactly 2 packages are late on a day.
2. $X$ is the number of late packages on a randomly selected day. (correct answer)
3. $X$ is the event that no packages are late.
4. $X$ is the set of probabilities associated with late packages.
5. $X$ is the long-run proportion of days with at most 1 late package.

Explanation: This problem tests the interpretation of discrete random variables within AP Statistics' framework of probability distributions. X represents the number of late packages on a randomly selected day, a discrete random variable with possible values like 0, 1, 2, etc., and their probabilities in the table. The correct choice, B, states that X is the number of late packages, which matches the question's definition precisely. Choice A distracts by confusing X with a probability for a specific outcome, but X is the variable itself, not a probability. In a mini-lesson, discrete random variables model countable outcomes, and their distributions allow us to calculate probabilities for events, such as the chance of more than a certain number of late packages.

Question 10

1. $X$ is the number of muffins sold in a randomly selected hour. (correct answer)
2. $X$ is the probability that the café sells muffins in a randomly selected hour.
3. $X$ is the event that exactly 3 muffins are sold in an hour.
4. $X$ is the list of possible hours in the day when muffins can be sold.
5. $X$ is the probability distribution of muffin sales for the entire day.

Explanation: This question assesses recognition of discrete random variables in a business context. The random variable X represents the number of muffins sold in a randomly selected hour, which is a countable quantity that varies randomly. Choice B incorrectly interprets X as a probability value rather than a count. Choice C mistakes X for a specific event (selling exactly 3 muffins) rather than a variable that can take different values. Choice D confuses X with a list of time periods rather than a numerical quantity. Choice E misidentifies X as the entire probability distribution rather than the variable itself. A discrete random variable like X takes on countable values (0, 1, 2, 3... muffins) and has an associated probability distribution that tells us how likely each value is.

Question 11

1. $X$ is the set of traffic lights on the route to work.
2. $X$ is the probability that the commuter encounters no red lights on the route.
3. $X$ is the number of red lights encountered on the route on a randomly selected day. (correct answer)
4. $X$ is the event that the commuter encounters at least 3 red lights.
5. $X$ is the probability distribution of all commuters' red-light counts in the city.

Explanation: This question evaluates interpretation of random variables in transportation contexts. The random variable X counts the number of red lights encountered on a specific route on a randomly selected day. Choice A incorrectly identifies X as a set of traffic lights rather than a numerical count. Choice B mistakes X for a probability value rather than a count of red lights. Choice D confuses X with a specific event rather than a variable that can take different values. Choice E misinterprets the scope, referring to all commuters rather than this specific commuter's experience. A random variable assigns numerical values to outcomes - here, we randomly select a day and count the red lights encountered, making X a discrete random variable.

Question 12

1. $Y$ is the probability that a player opens the app on a given day.
2. $Y$ is the event that the player receives 20 bonus points.
3. $Y$ is the number of bonus points awarded in one daily reward opening. (correct answer)
4. $Y$ is the total bonus points awarded to all players in one day.
5. $Y$ is the list of possible outcomes (0, 5, 10, 20) regardless of probabilities.

Explanation: This question assesses comprehension of discrete random variables in a gaming context. The random variable Y represents "the number of bonus points awarded in one daily reward opening," taking values 0, 5, 10, or 20. Choice A incorrectly interprets Y as a probability of opening the app, not the points awarded. Choice B mistakes Y for a single event (receiving exactly 20 points) rather than a variable with multiple possible outcomes. Choice D confuses the individual player's reward with aggregate totals across all players. Choice E describes Y as merely a list of outcomes without recognizing it as a random variable with associated probabilities. A discrete random variable assigns numerical values to outcomes of a random process—here, the points awarded when a single player opens their daily reward.

Question 13

1. A variable whose values form a countable set of numbers, where each value is associated with a probability. (correct answer)
2. A variable in an experiment whose value is measured to determine the effect of a treatment.
3. A numerical characteristic of a sample that is calculated from data, such as the sample mean.
4. A variable that can take on any value within a given range or interval, measured with a certain precision.

Explanation: A discrete random variable is a variable that can take on a finite or countably infinite number of distinct values. Its probability distribution assigns a probability to each possible value. Choice B describes a response variable. Choice C describes a statistic. Choice D describes a continuous random variable.

Question 14

1. The number of heads obtained in four flips of a fair coin.
2. The number of cars that pass through a toll booth in one hour.
3. The amount of time it takes a student to complete a homework assignment. (correct answer)
4. The number of correct answers on a 10-question multiple-choice test.

Explanation: A discrete random variable has a countable number of possible values (e.g., integers). A continuous random variable can take any value in an interval. The number of heads, cars, and correct answers are all countable integers, making them discrete. The amount of time can be any value in an interval (e.g., 20.5 minutes, 20.51 minutes), making it continuous.

Question 15

1. $X$ is the probability that a battery is defective.
2. $X$ is the event that the pack contains at least 1 defective battery.
3. $X$ is the number of defective batteries in a randomly selected pack of 6 batteries. (correct answer)
4. $X$ is the set of all possible packs of 6 batteries.
5. $X$ is the probability distribution for the number of nondefective batteries.

Explanation: This question tests understanding of random variables in manufacturing quality control. The random variable X represents the count of defective batteries found in a randomly selected pack of 6 batteries (choice C). X can take values 0, 1, 2, 3, 4, 5, or 6, representing the possible number of defective batteries. Choice A incorrectly describes X as a probability, choice B describes X as an event (having at least one defective battery), choice D confuses X with a set of battery packs, and choice E misidentifies X as relating to nondefective batteries. Random variables represent numerical outcomes we observe, not probabilities or events.

Question 16

1. $X$ is the probability that a driver delivers at least 1 package in 30 minutes.
2. $X$ is the number of packages delivered by a randomly selected driver in a randomly selected 30-minute interval. (correct answer)
3. $X$ is the event that exactly 2 packages are delivered in 30 minutes.
4. $X$ is the total number of packages delivered by the company in one day.
5. $X$ is the set of drivers who deliver packages in that interval.

Explanation: This question tests interpretation of random variables in logistics. The random variable X counts the number of packages delivered by a randomly selected driver in a randomly selected 30-minute interval. Choice A incorrectly describes X as a probability rather than a count. Choice C mistakes X for a specific event rather than a variable that can take multiple values. Choice D confuses the scope by referring to total daily packages rather than packages in a 30-minute interval. Choice E misidentifies X as a set of drivers rather than a numerical quantity. The key to understanding random variables is recognizing that they assign numbers to random outcomes - here, the outcome is selecting a driver and time interval, and the number assigned is the package count.

Question 17

1. $X$ is the probability that a randomly selected student participates in at least one club.
2. $X$ is the number of clubs a randomly selected student participates in. (correct answer)
3. $X$ is the set of all clubs offered at the school.
4. $X$ is the probability distribution of club participation for all students combined.
5. $X$ is the event that a student participates in exactly 2 clubs.

Explanation: This question tests understanding of what a random variable represents in probability. A random variable is a function that assigns numerical values to outcomes of a random phenomenon. In this context, X represents the number of clubs a randomly selected student participates in, making it a discrete random variable that can take values like 0, 1, 2, etc. Choice A incorrectly describes X as a probability rather than a count. Choice C wrongly identifies X as a set of clubs rather than a numerical quantity. Choice D confuses the random variable with its probability distribution. Choice E describes X as a specific event rather than a variable. Remember that a random variable is a numerical quantity whose value depends on the outcome of a random phenomenon, not a probability or event itself.

Question 18

1. X: 0, 1, 2, 3; P(X=x): 0.6, 0.2, 0.1, 0.2
2. X: 0, 1, 2, 3; P(X=x): 0.7, 0.2, 0.1, 0.0 (correct answer)
3. X: 0, 1, 2, 3; P(X=x): 0.6, 0.3, -0.1, 0.2
4. X: 0, 1, 2, 3; P(X=x): 0.5, 0.2, 0.2, 0.0

Explanation: A valid probability distribution requires two conditions: (1) all probabilities must be between 0 and 1, inclusive, and (2) the sum of all probabilities must equal 1. Choice B is the only option that satisfies both conditions: $0.7 + 0.2 + 0.1 + 0.0 = 1.0$. Choice A has probabilities that sum to 1.1. Choice C includes a negative probability. Choice D has probabilities that sum to 0.9.

Question 19

1. 0.15
2. 0.25
3. 0.30 (correct answer)
4. 0.70

Explanation: The sum of the probabilities in a valid probability distribution must be 1. Therefore, $0.15 + k + 0.35 + 0.20 = 1$. Combining the known probabilities gives $0.70 + k = 1$. Solving for k yields $k = 1 - 0.70 = 0.30$.

Question 20

1. For a large number of randomly selected milk cartons, the proportion that are no longer fresh on the expiration date itself is expected to be approximately 0.60. (correct answer)
2. For any randomly selected milk carton, it is more likely than not to have a freshness value of 0.60 days after expiration, indicating it is not fresh.
3. In any random sample of 10 cartons of milk, it is certain that exactly 6 of them will no longer be fresh on the expiration date.
4. The probability that the average number of fresh days for a large sample of milk cartons will be 0.60 is very high since this is the most likely outcome.

Explanation: The probability of an event can be interpreted as the long-run relative frequency of that event. Therefore, if many cartons of milk are observed, we expect the proportion that are not fresh on the expiration date ($X=0$) to be close to 0.60. Choice B confuses the probability with the value of the random variable. Choice C uses deterministic language ('certain that exactly 6') which is incorrect for a random process. Choice D incorrectly discusses the average number of fresh days, not the probability of a single outcome.