The Geometric Distribution
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AP Statistics › The Geometric Distribution
A student practices free throws and records each attempt as “made” or “missed.” The student keeps shooting until making the first free throw, then stops. Assume the chance of making any given free throw stays the same from shot to shot and each shot outcome is independent. The student wants to model the number of shots taken until the first made free throw. Why is a geometric model appropriate?
Because the student stops after the first success and each shot is an independent Bernoulli trial with constant success probability
Because it applies only when there are more than two possible outcomes on each trial
Because it models the number of successes in a fixed number of shots
Because it requires sampling without replacement so that the probability changes each shot
Because the number of shots is fixed ahead of time and the number made is random
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.
A researcher administers a short survey to randomly selected people walking through a mall until the first person agrees to participate. Each person either “agrees” or “declines,” and the researcher stops after the first agreement. Assume the probability a person agrees is approximately constant and each selection is independent. The researcher wants to model the number of people asked until the first agreement. Why is a geometric model appropriate?
Because the number of people asked is fixed and the number agreeing is random
Because the probability of agreement must decrease as more people are asked
Because the situation involves sampling without replacement from a small, finite population so probabilities change each time
Because the researcher is counting how many people agree out of a fixed sample size
Because each trial has two outcomes with constant probability of success, trials are independent, and the variable counts trials until the first success
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.
A website shows an advertisement to a user repeatedly until the user clicks it for the first time. Each display results in either “click” or “no click.” Assume the probability of a click is constant from display to display and outcomes are independent. The analyst wants to model the number of displays until the first click. Why is a geometric model appropriate?
Because it is designed for outcomes with more than two categories per trial
Because it applies when the probability of a click increases with each additional display
Because it models the number of clicks in a fixed number of ad displays
Because it requires the number of displays to be fixed in advance
Because it models independent Bernoulli trials with constant success probability and counts trials until the first success
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.
A nurse repeatedly checks a patient’s blood pressure each hour until the first time the reading falls within a target range. Each hour’s reading is classified as either “in range” or “not in range,” and the nurse stops once the first “in range” reading occurs. Assume the probability a reading is in range is constant from hour to hour and readings are independent. The nurse wants to model the number of hours checked until the first in-range reading. Why is a geometric model appropriate?
Because the number of hours is fixed and only the probability of success varies
Because it models trials until the first success under independent Bernoulli trials with constant success probability
Because it models the number of in-range readings in a fixed number of hours
Because it is appropriate whenever the mean of the data is larger than the standard deviation
Because the probability of an in-range reading must change each hour due to treatment
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.
A call center representative makes sales calls until getting the first customer who agrees to buy a service plan. Each call results in either “sale” or “no sale.” The representative assumes the probability of a sale is the same on each call and that calls are independent. The representative wants to model the number of calls made until the first sale. Why is a geometric model appropriate?
Because the probability of a sale changes after each call as the representative gains experience
Because it counts the number of trials until the first success under independent Bernoulli trials with constant success probability
Because it counts the number of sales in a fixed set of $n$ calls
Because it requires selecting customers without replacement from a small, known list
Because it requires a fixed number of calls and a variable number of successes
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.
A machine fills bottles, and a technician inspects bottles one at a time until finding the first bottle that is underfilled. Each inspection results in either “underfilled” or “not underfilled.” Assume the probability a bottle is underfilled is constant over time and inspections are independent. The technician wants to model the number of bottles inspected until the first underfilled bottle. Why is a geometric model appropriate?
Because it models the number of inspections until the first success with independent trials and constant probability of success
Because it requires a fixed number of inspections and a varying number of underfilled bottles
Because it models the number of underfilled bottles in a fixed set of $n$ inspected bottles
Because the outcomes are measured on a continuous scale rather than success/failure
Because the probability of an underfilled bottle changes deterministically after each inspection
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.
A nurse attempts to start an IV line and repeats attempts until achieving the first successful insertion. Assume each attempt is independent and the probability of success is constant for this nurse in this setting. The nurse records the number of attempts until the first success. Why is a geometric model appropriate?
Because attempts are not independent since each attempt changes the probability of success in a known way.
Because the random variable is the number of attempts until the first success with constant success probability.
Because the nurse is counting the number of failures in a fixed set of attempts.
Because the nurse stops after exactly 5 attempts regardless of success.
Because the number of attempts is fixed and the nurse counts the number of successful insertions.
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.
A gamer opens virtual packs one at a time until obtaining the first rare item. Assume each pack opening is independent and the probability a pack contains a rare item is constant. The gamer records the number of packs opened until the first rare item. Why is a geometric model appropriate?
Because the gamer opens a fixed number of packs and counts how many rare items appear.
Because the process continues until the first success and each trial has the same probability of success.
Because the probability of a rare item increases with each pack opened due to a guaranteed reward system.
Because multiple rare items must be obtained before stopping.
Because the number of packs opened is fixed in advance.
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.
A quality-control inspector tests light bulbs from a large shipment one at a time until finding the first defective bulb. Each bulb is classified as defective or not defective, and the chance a bulb is defective is assumed to be constant from test to test. The inspector records the number of bulbs tested until the first defective bulb is found. Why is a geometric model appropriate for this situation?
Because the probability of a defective bulb changes after each bulb is tested.
Because the inspector is interested in the number of defective bulbs in a fixed sample of $n$ bulbs.
Because the trials continue until the first success, with independent trials and constant probability of success.
Because the inspector will always test exactly the same number of bulbs each time.
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.
A driver approaches traffic lights along a route and notes each light as green or not green upon arrival. The driver continues until encountering the first red light. Assume each light encountered is independent and the probability a light is red upon arrival is constant. The driver records the number of lights encountered until the first red light. Why is a geometric model appropriate?
Because the driver plans to observe exactly $n$ lights and count how many are red.
Because the driver will always encounter the same number of lights on every trip.
Because a red light can occur more than once in the same trial.
Because the driver is counting trials until the first success (a red light) with constant probability each trial.
Because the probability a light is red depends on how many green lights have occurred previously.
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.