AP Statistics Quiz: The Geometric Distribution

What this quiz covers

This quiz focuses on The Geometric Distribution, 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. Because it models the number of successes in a fixed number of shots
2. Because it requires sampling without replacement so that the probability changes each shot
3. Because the student stops after the first success and each shot is an independent Bernoulli trial with constant success probability (correct answer)
4. Because the number of shots is fixed ahead of time and the number made is random
5. Because it applies only when there are more than two possible outcomes on each trial

Explanation: In AP Statistics, this question explores the geometric distribution, ideal for scenarios where we count trials until the first success in independent Bernoulli trials with fixed success probability. Each free throw is a Bernoulli trial with 'success' as making the shot, and the student stops at the first success, aligning with the geometric setup. The 'until first success' framework captures the random number of shots, including the successful one. Distractor A mistakenly applies the binomial model, which counts successes in a fixed number of trials, but here the number of trials is variable. Mini-lesson: The geometric distribution describes the probability of needing exactly k trials for the first success, with P(X=k) = (1-p)^{k-1} p, requiring independence and constant p, as assumed in the independent shots with unchanging probability.

Question 2

1. Because it models the number of mountain-lion photos in a fixed set of $n$ photos
2. Because it requires a fixed number of photos reviewed and counts how many contain mountain lions
3. Because it assumes the probability of success decreases as more photos are reviewed
4. Because it models the number of trials until the first success with independent Bernoulli trials and constant success probability (correct answer)
5. Because it is used when outcomes are normally distributed

Explanation: This AP Statistics question evaluates knowledge of the geometric distribution for counting trials to the first success in independent Bernoulli trials with fixed p. Reviewing each photo is a trial with 'success' as containing a mountain lion, stopping at the first success. The 'until first success' is ideal as reviews continue until the first such photo. Distractor A implies binomial with fixed n, incorrect here. Mini-lesson: Geometric model: P(X=k) = (1-p)^{k-1} p for k trials to first success; assumes independent trials, constant p, matching the independent photos with constant probability.

Question 3

1. Because the researcher is counting how many people agree out of a fixed sample size
2. Because the number of people asked is fixed and the number agreeing is random
3. Because each trial has two outcomes with constant probability of success, trials are independent, and the variable counts trials until the first success (correct answer)
4. Because the probability of agreement must decrease as more people are asked
5. Because the situation involves sampling without replacement from a small, finite population so probabilities change each time

Explanation: This AP Statistics item examines the geometric distribution, suitable for counting independent Bernoulli trials until the first success with unchanging p. Approaching each person is a trial with 'success' as agreement, stopping at the first success. The 'until first success' model applies since the researcher persists until the first agreement, with variable trials. A distractor like A confuses it with binomial by assuming a fixed sample. Mini-lesson: Geometric distribution gives P(X=k) = (1-p)^{k-1} p for k=1,2,...; it requires binary outcomes, independence, and constant p, as in the independent selections with approximate constant agreement probability.

Question 4

1. Because it models the number of clicks in a fixed number of ad displays
2. Because it requires the number of displays to be fixed in advance
3. Because it applies when the probability of a click increases with each additional display
4. Because it models independent Bernoulli trials with constant success probability and counts trials until the first success (correct answer)
5. Because it is designed for outcomes with more than two categories per trial

Explanation: Focusing on the geometric distribution in AP Statistics, this question highlights its use for the number of trials until the first success in independent Bernoulli trials with constant probability. Each ad display is a Bernoulli trial with 'success' as a click, stopping at the first success. The 'until first success' aspect is evident as displays continue until the initial click, making the count random. Distractor B incorrectly suggests a fixed number of displays, which would suit binomial, not geometric. Mini-lesson: The geometric PMF is P(X=k) = (1-p)^{k-1} p, applicable when trials are independent, p is constant, and we're interested in the trial of the first success; here, independent displays with fixed click probability satisfy this.

Question 5

1. Because it models the number of in-range readings in a fixed number of hours
2. Because it models trials until the first success under independent Bernoulli trials with constant success probability (correct answer)
3. Because the number of hours is fixed and only the probability of success varies
4. Because the probability of an in-range reading must change each hour due to treatment
5. Because it is appropriate whenever the mean of the data is larger than the standard deviation

Explanation: Exploring the geometric distribution in AP Statistics, this question justifies its use for trials until first success in independent Bernoulli trials with constant probability. Each hourly check is a trial with 'success' as in-range, stopping at the first success. The 'until first success' framework matches the ongoing checks until the initial in-range reading. Distractor A suggests binomial for fixed trials, which doesn't apply. Mini-lesson: The geometric distribution's PMF is P(X=k) = (1-p)^{k-1} p, requiring independence and constant p, fulfilled by the independent readings with fixed in-range probability.

Question 6

1. Because the probability of a sale changes after each call as the representative gains experience
2. Because it counts the number of trials until the first success under independent Bernoulli trials with constant success probability (correct answer)
3. Because it counts the number of sales in a fixed set of $n$ calls
4. Because it requires a fixed number of calls and a variable number of successes
5. Because it requires selecting customers without replacement from a small, known list

Explanation: This AP Statistics question tests recognition of the geometric distribution for modeling trials until the first success in independent Bernoulli trials with constant p. Each sales call is a Bernoulli trial where 'success' is making a sale, and the representative continues until the first success. The 'until first success' structure fits because the process halts at the first sale, with the random variable being the number of calls made. Choice C is a distractor that describes the binomial distribution instead, which uses a fixed number of trials. For a mini-lesson: Geometric distribution probabilities are calculated as P(X=k) = (1-p)^{k-1} p for k trials until first success; it assumes trials are independent and p is constant, matching the independent calls with fixed sale probability here.

Question 7

1. Because it models the number of underfilled bottles in a fixed set of $n$ inspected bottles
2. Because it models the number of inspections until the first success with independent trials and constant probability of success (correct answer)
3. Because it requires a fixed number of inspections and a varying number of underfilled bottles
4. Because the probability of an underfilled bottle changes deterministically after each inspection
5. Because the outcomes are measured on a continuous scale rather than success/failure

Explanation: In AP Statistics, this question addresses the geometric distribution for scenarios involving trials until the first success in independent Bernoulli settings with fixed p. Inspecting each bottle is a Bernoulli trial with 'success' as underfilled, continuing until the first such bottle. The 'until first success' structure is appropriate as inspections stop at the initial underfilled one, randomizing the count. Distractor A describes binomial counting in fixed trials, not applicable here. Mini-lesson: The geometric model uses P(X=k) = (1-p)^{k-1} p, assuming independent trials and constant success probability, which holds for the independent inspections with constant underfill probability.

Question 8

1. Because the number of attempts is fixed and the nurse counts the number of successful insertions.
2. Because attempts are not independent since each attempt changes the probability of success in a known way.
3. Because the random variable is the number of attempts until the first success with constant success probability. (correct answer)
4. Because the nurse stops after exactly 5 attempts regardless of success.
5. Because the nurse is counting the number of failures in a fixed set of attempts.

Explanation: This question tests recognition of geometric distribution in a medical procedure context. The geometric distribution is appropriate when counting trials until first success, with independent trials and constant probability of success. Here, the nurse attempts IV insertions until achieving the first successful insertion, each attempt is assumed independent, and the success probability is constant for this nurse in this setting. Choice C correctly identifies the key feature: counting attempts until first success with constant probability. Choice A describes a binomial distribution, choice B incorrectly claims dependence between attempts, choice D implies a fixed number of attempts, and choice E describes counting failures in fixed trials.

Question 9

1. Because the gamer opens a fixed number of packs and counts how many rare items appear.
2. Because the probability of a rare item increases with each pack opened due to a guaranteed reward system.
3. Because the process continues until the first success and each trial has the same probability of success. (correct answer)
4. Because the number of packs opened is fixed in advance.
5. Because multiple rare items must be obtained before stopping.

Explanation: This question assesses understanding of geometric distribution in gaming contexts. The geometric distribution models the number of trials needed until the first success occurs, where trials are independent with constant probability. In this scenario, the gamer opens packs until obtaining the first rare item (success), each pack opening is independent, and the probability of getting a rare item is constant. Choice C correctly identifies the key features: continuing until first success with constant probability. Choice A describes a binomial situation, choice B violates the constant probability assumption with a pity system, choice D implies predetermined trials, and choice E requires multiple successes rather than just the first.

Question 10

1. Because the inspector is interested in the number of defective bulbs in a fixed sample of $n$ bulbs.
2. Because the probability of a defective bulb changes after each bulb is tested.
3. Because the trials continue until the first success, with independent trials and constant probability of success. (correct answer)
4. Because the inspector will always test exactly the same number of bulbs each time.
5. Because more than one defective bulb must be found before stopping.

Explanation: This question tests understanding of when to use the geometric distribution. The geometric distribution models the number of trials needed until the first success occurs, where each trial is independent with constant probability of success. In this scenario, the inspector tests bulbs one by one until finding the first defective bulb (the 'success'), each test is independent, and the probability of finding a defective bulb remains constant. Choice C correctly identifies all three key features: trials continue until first success, trials are independent, and probability is constant. Choice A incorrectly describes a binomial situation (fixed n, counting successes), while choices B, D, and E describe situations that violate the geometric model's assumptions.

Question 11

1. Because the driver plans to observe exactly $n$ lights and count how many are red.
2. Because the driver is counting trials until the first success (a red light) with constant probability each trial. (correct answer)
3. Because the probability a light is red depends on how many green lights have occurred previously.
4. Because the driver will always encounter the same number of lights on every trip.
5. Because a red light can occur more than once in the same trial.

Explanation: This question tests recognition of geometric distribution in a transportation scenario. The geometric distribution applies when we count independent trials with constant probability until the first success occurs. Here, the driver encounters lights until hitting the first red light (success), each light is independent, and the probability of encountering a red light is constant. Choice B correctly identifies the essential feature: counting trials until first success (red light) with constant probability. Choice A describes a binomial distribution with fixed observations, choice C violates the independence assumption, choice D incorrectly focuses on fixed trial count, and choice E doesn't make sense as only one outcome occurs per light.

Question 12

1. Because the number of trials is fixed and the number correct is counted.
2. Because each trial results in success/failure and the variable is the number of trials until the first success. (correct answer)
3. Because the probability of success depends on how many guesses have already been made.
4. Because the student must get exactly 3 correct answers before stopping.
5. Because the student is sampling without replacement from a small set of questions.

Explanation: This question assesses understanding of geometric distribution in an educational testing context. The geometric distribution models the number of trials needed until the first success, where trials are independent with constant probability of success. In this scenario, the student guesses until getting the first correct answer (success), each guess is independent, and the probability of a correct guess remains constant. Choice B correctly identifies the key features: success/failure outcomes and counting trials until first success. Choice A describes a binomial distribution, choice C violates the constant probability assumption, choice D requires multiple successes, and choice E violates the independence assumption by suggesting sampling without replacement.

Question 13

1. Because the count of answered calls is measured in a fixed set of $n$ dials.
2. Because the number of dials is fixed and known before dialing begins.
3. Because each dial has two outcomes and the process stops at the first success with constant success probability. (correct answer)
4. Because the probability of an answer changes every time a new number is dialed.
5. Because the supervisor is interested in the number of failures after exactly 20 dials.

Explanation: This question tests recognition of geometric distribution in a call center scenario. The geometric distribution is appropriate when we have repeated independent trials with constant probability of success, continuing until the first success occurs. Here, each dial is an independent trial with two outcomes (answered or not), the probability of an answer is constant, and dialing continues until the first answered call. Choice C correctly identifies all key features: binary outcomes, stopping at first success, and constant success probability. Choice A describes counting successes in fixed trials (binomial), choice B incorrectly focuses on fixed trials, choice D violates the constant probability assumption, and choice E describes a different type of problem entirely.

Question 14

1. Because the company is counting the number of clicks in a fixed number of displays.
2. Because the probability of a click must decrease as the visitor becomes annoyed.
3. Because the outcome of each display can be categorized as click or no click, and the count is displays until the first click. (correct answer)
4. Because the company predetermines the number of displays and stops after that many no matter what.
5. Because more than one click can occur on the same display.

Explanation: This question examines understanding of geometric distribution in a digital marketing context. The geometric distribution models situations where we count trials until the first success, with each trial having two outcomes (success/failure) and constant probability of success. In this scenario, each ad display is a trial with two outcomes (click or no click), displays continue until the first click occurs, and the probability of clicking remains constant. Choice C correctly identifies these features: binary outcomes and counting displays until first click. Choice A describes a binomial situation, choice B incorrectly suggests decreasing probability, choice D implies a fixed number of trials, and choice E is impossible since only one click can occur per display.

Question 15

1. Because the number of trials is fixed in advance and the number of successes is counted.
2. Because the probability of success is constant and the count is trials until the first success. (correct answer)
3. Because the number of successes can be more than one in the same trial.
4. Because the free throws are not independent due to practice effects within the session.
5. Because the player always takes exactly 10 shots each practice session.

Explanation: This question assesses recognition of geometric distribution characteristics in a sports context. The geometric distribution applies when counting trials until the first success, with each trial being independent and having constant probability p. Here, the player shoots until making the first basket (success), each shot is independent, and the probability of making a shot remains constant at p. Choice B correctly identifies both key features: constant probability and counting trials until first success. Choice A describes a binomial distribution (fixed trials, counting successes), choice C doesn't apply to this scenario, choice D violates the independence assumption, and choice E describes a fixed number of trials rather than the 'until first success' structure.

Question 16

1. Because it models the number of critical errors in a fixed number of runs
2. Because it requires the number of runs to be fixed in advance
3. Because it models the number of runs until the first success when each run is an independent Bernoulli trial with constant success probability (correct answer)
4. Because it applies only when the probability of a critical error changes from run to run
5. Because it is the same as a binomial model with a fixed $n$ and varying $p$

Explanation: In AP Statistics, this question confirms the geometric distribution's appropriateness for the number of trials until first success in independent Bernoulli trials with constant p. Each test run is a Bernoulli trial with 'success' as critical error, stopping at the first. The 'until first success' structure applies since runs persist until the initial error. Choice A distracts by describing binomial for fixed runs. Mini-lesson: Geometric PMF: P(X=k) = (1-p)^{k-1} p; it needs binary outcomes, independence, constant p, as in the independent runs with fixed error probability.

Question 17

1. Because the inspector is counting the number of defective bulbs in a fixed sample of $n$ tested bulbs
2. Because each test is a Bernoulli trial with constant probability of "defective," trials are independent, and the variable counts trials until the first success (correct answer)
3. Because the probability of a defective bulb increases with each test as the inspector gets closer to finding one
4. Because the number of bulbs tested is fixed in advance and the number of defectives varies
5. Because the situation involves selecting without replacement from a small population, so probabilities change each trial

Explanation: This question assesses understanding of the geometric distribution in AP Statistics, which models the number of trials needed until the first success in a series of independent Bernoulli trials with constant success probability. Here, each bulb test is a Bernoulli trial where 'success' is finding a defective bulb, and the inspector continues until the first success, making the geometric model fitting. The 'until first success' structure is key because the process stops upon the first defective, and the random variable is the count of trials including that success. A common distractor is choice A, which confuses this with the binomial distribution by implying a fixed sample size, but here the number of tests is random, not fixed. In a mini-lesson on the geometric distribution: it applies to sequences of independent trials each with two outcomes (success or failure), constant p, and focuses on the trial number of the first success; the probability mass function is P(X=k) = (1-p)^{k-1} p for k=1,2,3,... The assumptions of independence and constant probability hold as stated, justifying the model.

Question 18

1. Because it counts the number of correct answers in a fixed number of attempted questions
2. Because the probability of a correct answer must increase after each incorrect attempt
3. Because the number of attempted questions is fixed in advance
4. Because each attempt is an independent Bernoulli trial with constant success probability and the random variable is trials until the first success (correct answer)
5. Because it applies only when sampling without replacement from a small set of questions

Explanation: This AP Statistics question probes the geometric distribution, which models the number of independent Bernoulli trials until the first success with constant p. Each question attempt is a trial with 'success' as correct, stopping at the first success. The 'until first success' fits because the student continues until the initial correct answer, with random attempts. Choice A is a distractor evoking binomial with fixed trials. Mini-lesson: Geometric probabilities are P(X=k) = (1-p)^{k-1} p; it demands binary outcomes per trial, independence, and fixed p, as seen in the independent guesses with same correctness probability.

Question 19

1. Because the number of flips is fixed in advance and the number of heads is counted.
2. Because the probability of heads changes based on previous flips.
3. Because there are two outcomes per trial and the random variable counts trials until the first success. (correct answer)
4. Because the coin is flipped exactly 10 times no matter what happens.
5. Because the researcher counts the number of heads in the first 10 flips.

Explanation: This question tests recognition of the geometric distribution in a classic probability experiment. The geometric distribution applies when counting independent trials until the first success, with constant probability of success on each trial. Here, the researcher flips until getting the first head (success), each flip is independent, and the probability of heads is constant. Choice C correctly identifies the essential features: binary outcomes per trial and counting trials until first success. Choice A describes a binomial distribution (fixed flips, counting heads), choice B incorrectly suggests changing probability, choice D implies fixed trials regardless of outcome, and choice E again describes a binomial situation.

Question 20

1. Because the technician inspects a fixed number of parts and counts how many fail.
2. Because the probability of failure is constant and the process stops when the first failure occurs. (correct answer)
3. Because the technician must observe exactly two failures to stop.
4. Because the number of inspected parts is fixed ahead of time.
5. Because the outcomes are measured on a continuous scale rather than success/failure.

Explanation: This question examines understanding of geometric distribution in quality control. The geometric distribution models situations where we perform independent trials with constant probability of success until the first success occurs. In this scenario, the technician inspects parts until finding the first failure (the 'success' in this context), each inspection is independent, and the probability of failure is constant. Choice B correctly identifies both key features: constant failure probability and stopping at first failure. Choice A describes a binomial situation, choice C requires multiple failures, choice D implies predetermined sample size, and choice E incorrectly suggests continuous rather than binary outcomes.