Random Variables and Probability Distributions
Help Questions
AP Statistics › Random Variables and Probability Distributions
A shipping company tracks the number of days a randomly selected package takes to arrive after it is shipped. Let $D$ be the discrete random variable representing the delivery time in days (1, 2, 3, or 4). The probability distribution of $D$ is shown in the table. Which statement correctly interprets the random variable?
$D$ is the event that a package takes 3 days to arrive.
$D$ is the set of probabilities associated with delivery times.
$D$ is the probability that a package arrives within 2 days.
$D$ is the number of days it takes a randomly selected package to be delivered.
$D$ is the number of packages delivered each day by the company.
Explanation
This question tests recognition of random variables in a shipping context. The random variable D represents "the number of days it takes a randomly selected package to be delivered," with possible values 1, 2, 3, or 4 days. Choice A incorrectly interprets D as a probability value rather than the delivery time itself. Choice C mistakes D for a specific event (taking exactly 3 days) rather than a variable that can take multiple values. Choice D confuses the delivery time of one package with the count of packages delivered daily. Choice E misunderstands D as the set of probabilities rather than the variable whose probabilities are described. A discrete random variable assigns numerical values to random outcomes—here, the number of days for one package's delivery.
A local library tracks how many books a randomly selected visitor checks out in a single visit. Let $B$ be the discrete random variable representing the number of books checked out (0, 1, 2, 3, or 4). The probability distribution of $B$ is shown in the table. Which statement correctly interprets the random variable?
$B$ is the number of books checked out by a randomly selected visitor in one visit.
$B$ is the total number of books in the library.
$B$ is the event that a visitor checks out exactly 0 books.
$B$ is the list of outcomes (0 through 4) rather than a random variable.
$B$ is the probability that a visitor checks out at least 1 book.
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.
A researcher randomly selects one household and records the number of pets the household owns. Let $P$ be the discrete random variable representing the number of pets (0, 1, 2, 3, or 4). The probability distribution of $P$ is given in the table. Which statement correctly interprets the random variable?
$P$ is the list of possible pet counts without considering their probabilities.
$P$ is the event that a household owns exactly 2 pets.
$P$ is the probability that a household owns pets.
$P$ is the number of pets owned by a randomly selected household.
$P$ is the total number of pets in the entire sample of households.
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.
A call center records the number of calls a randomly selected employee answers in the first hour of a shift. Let $N$ be the discrete random variable representing the number of calls answered (0, 1, 2, 3, or 4). The probability distribution of $N$ is shown in the table. Which statement correctly interprets the random variable?
$N$ is the number of calls answered in the first hour by a randomly selected employee.
$N$ is the probability that an employee answers at least 3 calls.
$N$ is the event that an employee answers exactly 4 calls.
$N$ is the probability distribution of calls for all employees combined.
$N$ is the total number of calls received by the call center in the first hour.
Explanation
This question assesses recognition of random variables in a workplace context. The random variable N represents "the number of calls answered in the first hour by a randomly selected employee," with possible values 0, 1, 2, 3, or 4. Choice B incorrectly interprets N as a probability of answering calls rather than the count itself. Choice C mistakes N for a specific event (answering exactly 4 calls) instead of a variable with multiple outcomes. Choice D confuses the individual employee's calls with the total received by the entire call center. Choice E misunderstands N as representing the combined distribution for all employees. A discrete random variable maps outcomes to numerical values—here, counting calls handled by one employee in their first hour.
A coffee shop tracks the number of pastries a randomly selected customer buys during a visit. Let $X$ be the number of pastries purchased. The probability distribution of $X$ is given in the table. Which statement correctly interprets the random variable $X$?
$X$ is the number of pastries purchased by a randomly selected customer during a visit.
$X$ is the event that a customer buys at least 1 pastry.
$X$ is the probability that a customer buys pastries.
$X$ is the list of possible outcomes for pastry purchases: $0,1,2,3$.
$X$ 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.
A student records the number of text messages they receive in a 10-minute interval. Let $X$ be the number of texts received in a randomly selected 10-minute interval. The probability distribution of $X$ is shown in the table. Which statement correctly interprets the random variable $X$?
$X$ is the number of texts received in a randomly selected 10-minute interval.
$X$ is the probability distribution table for texts received.
$X$ is the event that at least 2 texts are received in 10 minutes.
$X$ is the probability of receiving 0 texts in 10 minutes.
$X$ is the set of outcomes ${0,1,2,3,4}$.
Explanation
This question assesses the ability to identify a random variable from its context. A random variable represents a numerical quantity that varies randomly - in this case, the count of text messages received in a 10-minute interval (choice C). The random variable X takes values like 0, 1, 2, 3, or 4 texts, each with associated probabilities. Choice A incorrectly identifies X as a single probability value, choice B describes X as an event, choice D confuses X with the entire probability distribution, and choice E describes X as a set of outcomes rather than the variable itself. A random variable is the numerical outcome we're measuring, not the probabilities or events associated with it.
A shipping company counts the number of damaged items in a randomly selected box of 5 items. Let $X$ be the number of damaged items in the box. The probability distribution of $X$ is given in the table. Which statement correctly interprets the random variable $X$?
$X$ is the event that the box has no damaged items.
$X$ is the probability that the box has exactly 2 damaged items.
$X$ is the list of probabilities in the distribution table.
$X$ is the probability that an item is damaged.
$X$ is the number of damaged items in a randomly selected box of 5 items.
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.
A delivery company tracks the number of packages that arrive late to a particular office on a given day. Let the discrete random variable $X$ be the number of late packages on a randomly selected day. The probability distribution of $X$ is shown in the table. Which statement correctly interprets the random variable $X$?
$X$ is the long-run proportion of days with at most 1 late package.
$X$ is the set of probabilities associated with late packages.
$X$ is the number of late packages on a randomly selected day.
$X$ is the event that no packages are late.
$X$ is the probability that exactly 2 packages are late on a day.
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.
A game uses a spinner that awards points. Let $X$ be the number of points won on one spin. The probability distribution of $X$ is given in the table. Which statement correctly interprets the random variable $X$?
$X$ is the event that the spinner awards at most 5 points.
$X$ is the number of points awarded on one spin of the spinner.
$X$ is the probability that the spinner lands on 10 points.
$X$ is the probability distribution table of points.
$X$ is the set of outcomes ${0,5,10,20}$.
Explanation
This question assesses understanding of random variables in game contexts. The random variable X represents the number of points awarded on a single spin of the spinner (choice C). X can take values 0, 5, 10, or 20 points, each with certain probabilities. Choice A incorrectly describes X as a probability, choice B describes X as an event (points ≤ 5), choice D confuses X with the probability distribution table itself, and choice E describes X as merely the set of possible outcomes rather than the variable representing the actual points won. Random variables represent numerical outcomes of random experiments, not probabilities or sets.
A movie theater tracks the number of times a randomly selected customer refills a drink during a movie. Let $X$ be the number of refills. The probability distribution of $X$ is shown in the table. Which statement correctly interprets the random variable $X$?
$X$ is the event that a customer refills at most once.
$X$ is the probability that a customer refills exactly once.
$X$ is the probability distribution of refill counts, not a random variable.
$X$ is the list of possible refill counts $0,1,2,3$.
$X$ is the number of drink refills made by a randomly selected customer during a movie.
Explanation
This question assesses recognition of random variables in customer behavior contexts. The random variable X represents the number of drink refills made by a randomly selected customer during a movie (choice B). X takes values 0, 1, 2, or 3, representing the count of refills. Choice A incorrectly identifies X as a probability value, choice C describes X as an event (refills ≤ 1), choice D describes X as merely a list of values rather than the variable itself, and choice E suggests X is the distribution rather than a random variable. Remember that a random variable assigns numerical values to outcomes of random phenomena.