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AP Statistics Practice Quiz: The Geometric Distribution

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

According to the definition provided, what does a geometric random variable measure?

All Questions (16)

According to the definition provided, what does a geometric random variable measure?

A) The total number of successes in a fixed number of trials.

B) The trial number on which the first success occurs.

C) The probability of success on a single trial.

D) The average number of successes in a sequence of trials.

Correct Answer: B

The provided content explicitly states that a geometric random variable 'gives the trial number on which the first success occurs in a sequence of independent trials.'

A student is rolling a standard six-sided die until they roll a 4. This scenario can be modeled by a geometric distribution. What constitutes a 'trial' in this context?

A) The student successfully rolling a 4.

B) The number of rolls it takes to get the first 4.

C) A single roll of the die.

D) The student not rolling a 4.

Correct Answer: C

In a geometric distribution, a 'trial' is a single, independent event in the sequence. In this case, each roll of the die is one trial.

For a geometric distribution with a probability of success 'p', what is the formula for the mean?

A) p

B) 1/p

C) sqrt(1-p)/p

D) 1-p

Correct Answer: B

The content states that 'The mean of a geometric random variable is 1/p'.

A basketball player has a 60% chance of making any given free throw. If X is the geometric random variable for the attempt number of her first successful shot, what is the probability that her first success occurs on the 3rd attempt?

A) 0.096

B) 0.144

C) 0.240

D) 0.600

Correct Answer: A

The geometric probability function calculates P(X=x). Here, p=0.6, 1-p=0.4, and x=3. The probability is (1-p)^(x-1) * p = (0.4)^(3-1) * 0.6 = (0.4)^2 * 0.6 = 0.16 * 0.6 = 0.096.

A quality control process finds that 5% of items are defective. An inspector checks items one by one until the first defective item is found. What is the expected number of items the inspector will check?

A) 5

B) 10

C) 20

D) 50

Correct Answer: C

The expected number of trials until the first success is the mean of the geometric distribution. The probability of success (finding a defective item) is p=0.05. The mean is 1/p = 1/0.05 = 20.

An airline has determined that the probability of a passenger being a 'no-show' for a flight is 0.10. Let X be the number of passengers checked in until the first 'no-show' is identified. What is the standard deviation of X?

A) 3.00

B) 9.49

C) 10.00

D) 90.00

Correct Answer: B

The standard deviation of a geometric random variable is sqrt(1-p)/p. Here, p=0.10. So, the standard deviation is sqrt(1-0.10)/0.10 = sqrt(0.9)/0.10 ≈ 0.9487 / 0.10 ≈ 9.49.

A salesperson has a 25% chance of making a sale with any potential customer. The number of customers they approach until their first sale follows a geometric distribution. The mean of this distribution is 4. How should this parameter be interpreted?

A) The salesperson is guaranteed to make a sale by the 4th customer.

B) The salesperson will most likely make a sale on the 4th attempt.

C) Over many, many sequences of attempts, the average number of customers approached to get the first sale is 4.

D) The probability of making a sale on the 4th attempt is higher than any other attempt.

Correct Answer: C

The mean of a random variable, also known as the expected value, represents the long-run average outcome. The content emphasizes that parameters should be interpreted in context. The mean of 4 indicates the long-run average, not a guarantee or the most likely single outcome.

In a game, the probability of winning a prize on any single attempt is 0.02. A player keeps trying until they win their first prize. The probability that the first win occurs on the 10th attempt is calculated to be approximately 0.0167. Which of the following is the correct interpretation of this probability?

A) There is a 1.67% chance that a player will win exactly 10 times.

B) If a player tries 10 times, they are 1.67% likely to win.

C) In a large number of repeated games, about 1.67% of players will experience their first win on their 10th attempt.

D) A player is expected to win their first prize after 1.67 attempts.

Correct Answer: C

The geometric probability function calculates the probability of the first success occurring on a specific trial 'x'. This value should be interpreted in context as the likelihood of that specific event happening. It represents the long-run proportion of times this specific outcome would occur.

The mean number of attempts for a scientist to achieve a successful reaction in an experiment is 5. Assuming the trials are independent and the probability of success is constant, what is the probability that the first successful reaction occurs on the 2nd attempt?

A) 0.16

B) 0.20

C) 0.25

D) 0.80

Correct Answer: A

First, use the mean to find the probability of success, p. Mean = 1/p, so 5 = 1/p, which means p = 1/5 = 0.2. Now, calculate the probability that the first success is on the 2nd trial (x=2). P(X=2) = (1-p)^(2-1) * p = (1-0.2)^1 * 0.2 = 0.8 * 0.2 = 0.16.

A computer program has a bug that causes it to crash with a probability of 0.12 on any given run. The runs are independent. Which expression correctly calculates the probability that the program runs successfully 4 times before crashing for the first time on the 5th run?

A) (0.12)^4 * (0.88)

B) (0.88)^4 * (0.12)

C) (0.12)^5

D) (0.88)^5

Correct Answer: B

This is a geometric setting where 'success' is the program crashing. The probability of success is p=0.12. We want to find the probability that the first success occurs on trial x=5. The geometric probability function is P(X=x) = (1-p)^(x-1) * p. Plugging in the values: P(X=5) = (1-0.12)^(5-1) * 0.12 = (0.88)^4 * 0.12.

The expected number of job interviews a person needs to get their first job offer is 8. What is the standard deviation of the number of interviews needed?

A) 2.65

B) 7.48

C) 8.00

D) 56.00

Correct Answer: B

First, find the probability of success, p, from the mean. Mean = 1/p = 8, so p = 1/8 = 0.125. Next, calculate the standard deviation using the formula sqrt(1-p)/p. SD = sqrt(1-0.125)/0.125 = sqrt(0.875)/0.125 ≈ 0.9354 / 0.125 ≈ 7.48.

Which of the following scenarios is best modeled by a geometric random variable?

A) Counting the number of red cars in a sample of 50 cars passing an intersection.

B) Recording the number of times a student must take a driving test until they pass.

C) Measuring the average height of students in a classroom.

D) Counting the number of aces drawn from a deck of 52 cards without replacement.

Correct Answer: B

A geometric random variable measures the number of trials until the first success. Option B fits this perfectly: each test is a trial, and we are counting them until the first 'pass' (success). Option A is binomial, C is not a count of trials, and D lacks independent trials because the cards are not replaced.

For a geometric distribution, if the probability of success 'p' is very small, what can be said about the mean and standard deviation?

A) Both the mean and standard deviation will be small.

B) The mean will be large, and the standard deviation will be small.

C) The mean will be small, and the standard deviation will be large.

D) Both the mean and standard deviation will be large.

Correct Answer: D

The mean is 1/p and the standard deviation is sqrt(1-p)/p. If p is a very small number (close to 0), then 1/p will be a very large number, making the mean large. Similarly, 1-p will be close to 1, so sqrt(1-p) is also close to 1. The standard deviation, approximately 1/p, will also be very large.

A researcher is observing a rare celestial event that has a 0.005 probability of occurring each night. The researcher will observe each night until the event is seen for the first time. What is the correct interpretation of the standard deviation for this distribution?

A) It is the number of nights the researcher must wait to be certain of seeing the event.

B) It is the most likely number of nights the researcher will have to wait.

C) It measures the typical variation or spread in the number of nights one would expect to wait for the first event, centered around the mean.

D) It is the probability that the researcher will not see the event on any given night.

Correct Answer: C

The standard deviation of a distribution measures the typical or average distance of the outcomes from the mean. In the context of this geometric distribution, it quantifies the variability in the waiting time for the first success (the first observed event).

A factory produces light bulbs, and 3% of them are defective. The bulbs are tested one by one. Let X be the number of the bulb that is the first to be found defective. The probability that the first defective bulb is the 20th one tested is P(X=20) = (0.97)^19 * (0.03). What does the (0.97)^19 part of this calculation represent?

A) The probability of finding 19 defective bulbs.

B) The probability of finding 19 non-defective bulbs in a row.

C) The probability that the 19th bulb is the first defective one.

D) The average number of non-defective bulbs found.

Correct Answer: B

In the geometric probability formula, (1-p)^(x-1) represents the probability of having x-1 consecutive failures before the first success. Here, p=0.03 (defective), so 1-p=0.97 (non-defective), and x=20. Thus, (0.97)^(20-1) or (0.97)^19 is the probability of the first 19 bulbs being non-defective (failures to find a defective one).

The standard deviation of a geometric random variable X is 12. What is the probability that the first success occurs on the second trial, P(X=2)?

A) 0.0052

B) 0.0763

C) 0.0833

D) 0.1146

Correct Answer: B

This requires working backward. SD = sqrt(1-p)/p = 12. Squaring both sides gives (1-p)/p^2 = 144. This leads to the quadratic equation 144p^2 + p - 1 = 0. Using the quadratic formula, p = [-1 ± sqrt(1^2 - 4*144*(-1))] / (2*144) = [-1 ± sqrt(577)] / 288. Since p must be positive, p ≈ (-1 + 24.02) / 288 ≈ 0.08. Now, calculate P(X=2) = (1-p)*p ≈ (1-0.08)*0.08 = 0.92 * 0.08 = 0.0736. The closest answer is 0.0763, which results from using a more precise value for p (≈0.0799).