Random Variables and Probability Distributions - AP Statistics

1. 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?
   Answer: $B$ is the number of books checked out by a randomly selected visitor in one visit. 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.
2. What is the value of the constant c?
   Answer: 1/10 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 $1/10$.
3. A bakery fills a randomly selected box with a small number of cookies as a promotional sample. Define a discrete random variable $C$ as the number of cookies in the box (1, 2, or 3). The probability distribution of $C$ is shown in a table. Which statement correctly interprets the random variable?
   Answer: $C$ is the number of cookies in a randomly selected promotional box. This question tests understanding of random variables in a promotional context. The random variable C is defined as "the number of cookies in a randomly selected promotional box," taking values 1, 2, or 3. Choice A incorrectly interprets C as a probability value rather than the cookie count itself. Choice C mistakes C for an event about whether the bakery gives away cookies, not the quantity in a box. Choice D confuses the contents of one box with the total production for the day. Choice E misunderstands C as a general set of events rather than a specific numerical variable. A random variable assigns numbers to outcomes of a random process—in this case, counting cookies in one promotional box.
4. A teacher records whether a randomly selected student is absent on a particular day. Define a discrete random variable $A$ as the number of days absent for that student during a 5-day week (0 through 5). The probability distribution of $A$ is shown in a table. Which statement correctly interprets the random variable?
   Answer: $A$ is the number of days absent in a 5-day week for a randomly selected student. This question evaluates understanding of random variables in an educational attendance context. The random variable A is defined as "the number of days absent for that student during a 5-day week," which can take integer values from 0 to 5. Choice B incorrectly interprets A as a probability for a specific day rather than a count of absences. Choice C mistakes A for a single event (being absent exactly 2 days) instead of recognizing it as a variable with multiple possible values. Choice D confuses the individual student's absences with the class total. Choice E misidentifies A as the probability table itself rather than the variable whose distribution the table describes. A random variable maps outcomes to numbers—in this case, counting the total days one randomly selected student is absent in a week.
5. What is the value of $P(X \le 2)$?
   Answer: 0.7 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 $P(X=0) + P(X=1) + P(X=2) = 0.1 + 0.2 + 0.4 = 0.7$.
6. A cafeteria manager randomly selects one customer and defines a discrete random variable $X$ as the number of pieces of fruit the customer buys (0, 1, 2, or 3). The probability distribution of $X$ is shown in a table. Which statement correctly interprets the random variable?
   Answer: $X$ is the number of pieces of fruit bought by a randomly selected customer. This question tests understanding of what a random variable represents in probability. The random variable X is defined as "the number of pieces of fruit the customer buys," which means X takes on numerical values (0, 1, 2, or 3) representing a count. Choice A incorrectly interprets X as a probability value itself, when X is actually the variable whose probabilities are given in the distribution table. Choice C mistakenly treats X as a specific event rather than a variable that can take multiple values. Choice D confuses the individual customer's purchase with the total for all customers. Choice E misunderstands X as the set of probabilities rather than the variable itself. A random variable is a function that assigns numerical values to the outcomes of a random phenomenon—in this case, the number of fruit pieces purchased by one randomly selected customer.
7. 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 a table. Which statement correctly interprets the random variable $X$?
   Answer: $X$ is the number of drink refills made by a randomly selected customer during a movie. 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.
8. A student flips a fair coin 3 times. Let $X$ be a discrete random variable defined as the number of heads obtained in the 3 flips. The probability distribution of $X$ is given in the table. Which statement correctly interprets the random variable?
   Answer: $X$ is the number of heads obtained in 3 flips of a fair coin. This question evaluates understanding of random variables in a classic probability experiment. The random variable X represents the number of heads obtained when flipping a fair coin 3 times - a count that can be 0, 1, 2, or 3. Choice C correctly identifies X as this numerical outcome. Students often confuse X with a single-flip probability (A), a specific event (B), the sample space of all possible sequences (D), or the distribution table (E). The key concept is that X counts something specific (heads) across the entire experiment (3 flips), not individual outcomes. This is a binomial random variable, where we count successes (heads) in a fixed number of independent trials.
9. A delivery company defines the discrete random variable $X$ as the number of packages delivered by a randomly selected driver during a randomly selected 30-minute interval. The probability distribution for $X$ is given in the table. Which statement correctly interprets the random variable?
   Answer: $X$ is the number of packages delivered by a randomly selected driver in a randomly selected 30-minute interval. 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.
10. Given that at least one employee was absent, what is the probability that more than three employees were absent?
    Answer: 0.167 This is a conditional probability problem: $P(X > 3 | X \ge 1)$. The formula is $P(A|B) = P(A \text{ and } B) / P(B)$. Here, A is $X > 3$ and B is $X \ge 1$. The event '$X > 3$ and $X \ge 1$' is simply $X > 3$. We have $P(X > 3) = P(X=4) = 0.15$. We also have $P(X \ge 1) = 1 - P(X=0) = 1 - 0.10 = 0.90$. Therefore, the conditional probability is $0.15 / 0.90 \approx 0.167$.
11. 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?
    Answer: $P$ is the number of pets owned by a randomly selected household. 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.
12. What is the probability that a randomly selected person has more than 1 but at most 3 successful attempts?
    Answer: 0.5 We need to calculate $P(1 < Z \le 3)$. Using the cumulative probabilities, this is equivalent to $P(Z \le 3) - P(Z \le 1) = 0.8 - 0.3 = 0.5$.
13. A delivery company records the number of packages delivered to a particular office building each day. Let $X$ be a discrete random variable defined as the number of packages delivered to the building on a randomly selected day. The probability distribution of $X$ is shown in a table. Which statement correctly interprets the random variable?
    Answer: $X$ is the number of packages delivered to the building on a randomly selected day. This question tests recognition of a random variable in a real-world delivery context. The random variable X represents the count of packages delivered on any given day - a numerical quantity that varies randomly. Choice B correctly identifies X as this number. Common errors include confusing X with a specific event like "exactly 12 packages" (A), a probability value (C), a set of probabilities (D), or the distribution table itself (E). The key principle is that a discrete random variable assigns numerical values to outcomes of a random process. Here, each day's delivery count is an outcome, and X represents that count, while the probability distribution would tell us how likely each count is.
14. 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 $X$?
    Answer: $X$ is the number of pastries purchased by a randomly selected customer during a visit. 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.
15. A student takes a 4-question multiple-choice quiz where each question is either correct or incorrect. Let the discrete random variable $X$ be the number of questions answered correctly on the quiz. The probability distribution of $X$ is shown in a table. Which statement correctly interprets the random variable $X$?
    Answer: $X$ is the number of questions answered correctly on the quiz. In AP Statistics, this question focuses on understanding random variables and their probability distributions, specifically identifying what a discrete random variable represents. Here, X is the number of questions answered correctly on the quiz, a discrete variable since it can take integer values from 0 to 4, with probabilities shown in the table. Choice C accurately describes X as the number of questions answered correctly, aligning with the given definition. Choice A is a distractor because it mistakes X for a specific probability rather than the variable counting correct answers. For a mini-lesson, a random variable is a numerical outcome of a random phenomenon; discrete ones are countable, like the number of successes in binomial settings, which this quiz resembles with fixed trials.
16. A school counselor randomly selects a student and records the discrete random variable $X$, the number of extracurricular clubs that student participates in. The probability distribution for $X$ is shown in the table. Which statement correctly interprets the random variable?
    Answer: $X$ is the number of clubs a randomly selected student participates in. 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.
17. A machine produces light bulbs, and a quality inspector counts the number of defective bulbs in a randomly selected box of 5 bulbs. Let the discrete random variable $X$ be the number of defective bulbs in the box. The probability distribution of $X$ is shown in a table. Which statement correctly interprets the random variable $X$?
    Answer: $X$ is the number of defective bulbs in a randomly selected box of 5 bulbs. This question from AP Statistics evaluates knowledge of discrete random variables and their probability distributions. X counts defective bulbs in a box of 5, discrete with values 0 to 5 and tabled probabilities. Choice C correctly identifies X as this number, consistent with the definition. Distractor A equates X to a probability, but X is the count. Mini-lesson: In quality control, discrete variables like X often follow hypergeometric or binomial distributions, aiding in defect rate predictions and process improvements.
18. 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 a table. Which statement correctly interprets the random variable $X$?
    Answer: $X$ is the number of texts received in a randomly selected 10-minute interval. 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.
19. A website tracks the number of items a customer places in their cart during a single visit. Let the discrete random variable $X$ be the number of items placed in the cart on a randomly selected visit. The probability distribution of $X$ is shown in a table. Which statement correctly interprets the random variable $X$?
    Answer: $X$ is the number of items placed in the cart during a single visit. This AP Statistics question assesses understanding discrete random variables in probability contexts. X represents the number of items in the cart during a visit, discrete with values 0, 1, 2, 3 and associated probabilities. Choice C correctly interprets X as this number, aligning with the description. Distractor B mistakes X for a probability, whereas X is the count variable. Mini-lesson: Random variables turn qualitative data into quantitative; discrete ones like this enable e-commerce predictions, such as average items per visit via expected value.
20. Which of the following describes a continuous random variable rather than a discrete random variable?
    Answer: The amount of time it takes a student to complete a homework assignment. 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.

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