Random Variables and Probability Distributions
Help Questions
AP Statistics › Random Variables and Probability Distributions
Let X be a discrete random variable with a defined probability distribution. Which of the following statements about the probability distribution of X must be true?
The sum of the probabilities must equal 1, and each individual probability must be between 0 and 1, inclusive.
The values of X must be consecutive integers starting from 0 or 1, representing counts.
The distribution must be symmetric, with the mean equal to the median.
The expected value (mean) of X must be equal to one of the possible numerical values of X.
Explanation
This statement describes the two fundamental rules for any valid probability distribution. The sum of all probabilities for all possible outcomes must be exactly 1, and the probability of any single outcome must be a value from 0 to 1. Choice A is false, as distributions can be skewed. Choice C is false; the expected value is a weighted average and can be a value that the variable itself cannot take (e.g., 2.4 children). Choice D is false; the values can be any countable set of numbers, not necessarily consecutive integers (e.g., scores on a test).
What is the value of $$P(X \le 2)$$?
0.4
0.5
0.7
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 $$P(X=0) + P(X=1) + P(X=2) = 0.1 + 0.2 + 0.4 = 0.7$$.
What is the probability that a randomly selected watch has at least two defects?
0.04
0.06
0.07
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, $$P(X \ge 2) = P(X=2) + P(X=3) + P(X=4) = 0.04 + 0.02 + 0.01 = 0.07$$.
What is the probability that a randomly selected player scored more than 1 goal but no more than 4 goals?
0.55
0.70
0.85
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: $$P(Y=2) + P(Y=3) + P(Y=4) = 0.30 + 0.15 + 0.10 = 0.55$$.
What is the value of the constant c?
1/10
1/4
1
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 $$1/10$$.
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 number of packages delivered each day by the company.
$D$ 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 event that a package takes 3 days to arrive.
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 list of outcomes (0 through 4) rather than a random variable.
$B$ is the event that a visitor checks out exactly 0 books.
$B$ is the total number of books in the library.
$B$ 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 event that a household owns exactly 2 pets.
$P$ is the list of possible pet counts without considering their probabilities.
$P$ 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 probability distribution of calls for all employees combined.
$N$ 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 total number of calls received by the call center in the first hour.
$N$ is the event that an employee answers exactly 4 calls.
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 probability that a customer buys pastries.
$X$ is the probability that a customer buys exactly 2 pastries.
$X$ is the list of possible outcomes for pastry purchases: $0,1,2,3$.
$X$ is the event that a customer buys at least 1 pastry.
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.