Binomial and Geometric Practice Problems: Legacy Comparison
Work through a large legacy comparison bank without mistaking geometric distribution for current 2027 tested content; each problem begins with the model and event structure.
Binomial And Geometric Practice Problems: direct answer
In the second practice pass, classify each random variable before calculating. Keep the geometric waiting-time examples explicitly in the legacy bucket, and use the binomial examples to rehearse the fixed-n, two-outcome, independent-trials, constant-p conditions that remain current.
This page uses binomial and geometric practice problems as its single primary search focus. Every instructional block and every retained practice item is tied to that title intent rather than to a generic AP Statistics question-bank template.
Quick reference for Binomial and Geometric Practice Problems: Legacy Comparison
| Current priority | Binomial distribution remains current Unit 2 content. |
|---|---|
| Legacy topic | Geometric distribution was removed from the revised tested course. |
| First decision | Fixed n or stop at first success? |
| Second decision | Are trials independent with constant p? |
| Third decision | Translate exactly / at most / at least / more than before computing. |
Concept mastery: Binomial and Geometric Practice Problems: Legacy Comparison
How to use this legacy practice bank
Treat every problem as a model-selection exercise before calculating. Mark whether n is fixed, whether the process stops at first success, whether p is stable, and whether independence is defensible. Only then translate the requested event.
Current-course warning
The geometric distribution was removed from the revised 2026–27 AP Statistics tested content. These exercises are intentionally labeled legacy comparison practice so students can read older resources without confusing historical coverage with current exam priorities.
A solution should name the random variable
A complete response defines X in context, identifies the model, states the relevant parameter values, translates words such as at least or more than into an event, and interprets the result. Calculator output by itself does not show that the model matched the design.
Use errors diagnostically
If an answer is wrong, classify the reason: fixed-n versus stopping-time confusion, off-by-one boundary, incorrect complement, changing-p violation, dependence, or arithmetic. That error category is more useful than simply redoing the same calculation.
Translate the random variable before computing
Write X in words first. If X is “number of successes among n trials,” a binomial structure is possible. If X is “trial number of the first success,” the older geometric model is the natural comparison. This single sentence prevents most model-choice errors because it forces the stopping rule into view.
Separate current AP preparation from legacy review
The revised 2026–27 AP Statistics course removed geometric distribution from the tested content. These legacy exercises are therefore useful for interpreting older textbooks, released classroom materials, and conceptual contrasts, but current exam preparation should prioritize the binomial model and the current five-unit framework.
Check independence rather than assuming it
A two-outcome trial is not automatically independent. Sampling without replacement changes probabilities unless the sample is small relative to the population. Repeated trials can also be dependent when fatigue, learning, depletion, or feedback changes the chance of success.
Decode cumulative language carefully
For a fixed-n count, at most, fewer than, at least, and more than create different inclusive boundaries. For waiting-time problems, “more than k trials” means the first k trials are failures. Writing the event symbolically before using a calculator avoids off-by-one mistakes.
Use expected value as a long-run statement
Expected value is not a prediction that must occur in a single repetition. A binomial mean np is an average number of successes across many fixed-size repetitions. A legacy geometric mean 1/p is an average waiting time across many repeated first-success processes.
Recognize when neither model fits
Changing success probabilities, a random number of trials with more than one target success, multiple outcome categories that cannot be collapsed meaningfully, or strong dependence can invalidate both simple models. Model rejection is a legitimate statistical decision.
How to audit a completed legacy solution
After solving, verify the model statement, parameters, event boundary, complement logic, numerical value, and interpretation separately. A response can contain correct arithmetic but still be statistically wrong if it used a fixed-n count for a waiting-time process or treated a legacy geometric calculation as current tested content.
When to stop using the historical comparison
Once the stopping-rule contrast is secure, current 2027 preparation should return to binomial probability, sampling distributions, inference, data collection, and regression. The goal of this page is to decode older resources accurately, not to recreate the pre-revision syllabus.
Build an error log by decision type
Label each miss as model choice, condition failure, event translation, complement boundary, expected-value interpretation, or calculator execution. A decision-based error log reveals whether the same conceptual weakness is producing mistakes across superficially different scenarios.
Practice remediation: off-by-one errors
When a cumulative answer is wrong by one term, do not merely re-enter the calculator command. Rewrite the event as a set of allowable integer values. For a fixed-n count, “fewer than 4” includes 0,1,2,3; “at most 4” also includes 4. For a first-success waiting time, “more than 4 trials” describes four consecutive failures before any success. Writing the support and boundary makes the correction transferable to later problems.
Practice remediation: assumption errors
If the arithmetic is correct but the model is unjustified, identify which condition failed. Sampling a large fraction without replacement threatens independence and constant p; adaptive difficulty changes p; clustered units create dependence; stopping after a second success changes the waiting-time model. Recording the violated assumption is more valuable than changing a number because it teaches when the formula should not be used at all.
Practice remediation: interpretation errors
A probability answer must name the random variable and context. A binomial tail is a probability about the number of successes in a fixed batch. A historical waiting-time tail is a probability about how long it takes to reach first success. An expected value is a long-run average across repetitions of the complete process. These statements should not be swapped simply because the same p appears in both formulas.
Practice remediation: current-versus-legacy labeling
Every geometric item on this page is intentionally historical. When using an old review book, annotate such exercises as legacy so they do not distort a 2027 study schedule. The durable skill is the model-selection reasoning: define X, inspect n and the stopping rule, check independence and constant p, translate the event, and interpret the result. Current exam preparation should then emphasize the models and inference procedures that remain in the revised course.
Use this legacy bank as a controlled comparison exercise, not as evidence that geometric distribution has returned to the current AP Statistics tested framework.
Worked analysis for Binomial and Geometric Practice Problems: Legacy Comparison
Legacy practice clinic 1: fixed-count boundary
A retailer records exactly 16 sales contacts and asks for the probability of at least 4 conversions. The model decision is binomial; translate at least 4 as 1-P(X<=3) before using cumulative output.
Legacy practice clinic 2: waiting-time boundary
An archived exercise asks for more than 5 attempts before the first success. Under the historical geometric model, this means the first five attempts all fail, so the event can be evaluated as (1-p)^5.
Legacy practice clinic 3: changing probability
A shooter improves after every successful trial, so p changes. Even though outcomes are make/miss, neither a constant-p binomial count nor a simple geometric waiting-time model is appropriate.
Legacy practice clinic 4: without-replacement dependence
A small box is sampled without replacement and the sample is a large fraction of the box. The success probability changes materially after each draw, so a simple independent-trial model needs rejection or a different exact model.
Legacy practice clinic 5: exactly versus at most
For a fixed n problem, exactly 3 uses a single binomial probability; at most 3 accumulates probabilities for 0,1,2,3. The distinction is conceptual before it is computational.
Legacy practice clinic 6: first-success expectation
In a historical waiting-time problem with p=.20, the long-run mean trial number is 1/.20=5. This does not say the fifth trial must be the first success in any particular run.
Legacy practice clinic 7: binomial expectation
For n=40 and p=.30, the binomial mean is np=12 successes. It summarizes repeated 40-trial batches rather than predicting an integer result for every batch.
Legacy practice clinic 8: support check
A binomial count can take values 0 through n. A first-success waiting time takes positive integer values with no fixed upper endpoint. The support itself can reveal which random variable has been defined.
Legacy practice clinic 9: complement logic
At least one success in n independent trials is most efficiently computed as 1-P(no successes)=1-(1-p)^n. This is a binomial complement even though the formula resembles a run of failures.
Legacy practice clinic 10: legacy label
When an older AP review source assigns a geometric-distribution question, solve it only as historical enrichment and tag it as removed from current tested content so the study plan does not drift toward obsolete weighting.
Legacy practice clinic 11: two successes stopping rule
If trials continue until the second success, the process is not the simple first-success geometric model. The historical comparison page should not force every random stopping problem into geometric form.
Legacy practice clinic 12: finite campaign
A campaign sends exactly 75 messages and counts responses. Even if the analyst is especially interested in when the first response arrives, the stated response variable controls the model: count among 75 means binomial.
Legacy practice clinic 13: adaptive algorithm
An algorithm changes task difficulty after each failure, so the probability of success changes. Constant-p assumptions fail even if trials remain independent conditional on the difficulty level.
Legacy practice clinic 14: clustered trials
Responses from members of the same household are correlated. Treating them as independent Bernoulli trials understates dependence and can invalidate both simple models.
Legacy practice clinic 15: calculator audit
A calculator can produce a number for binompdf or a geometric formula even when the model is wrong. Always document the design check before accepting numerical output.
Legacy practice clinic 16: fixed n with zero successes
A binomial count allows X=0 because a fixed batch can finish without any successes. A first-success waiting time cannot equal 0; its support begins at trial 1.
Legacy practice clinic 17: probability one success
For a fixed n batch, exactly one success can occur in any of n positions, so the combination factor matters. In a first-success-on-trial-k calculation, the success position is already fixed at k.
Legacy practice clinic 18: event wording audit
Translate “fewer than 5 successes” as X≤4 before touching cumulative output. A one-word boundary error can change the event even when the model and parameters are correct.
Legacy practice clinic 19: model rejection practice
If outcomes are success/failure but p changes because the environment deteriorates after each trial, state that the simple model assumptions fail instead of forcing a familiar formula.
Legacy practice clinic 20: legacy textbook triage
When an older chapter mixes binomial and geometric sections, mark which exercises still match the revised course. Solve removed-topic items only when they strengthen model-selection understanding.
Legacy practice clinic 21: simulation check
A short simulation can reveal the different shapes: fixed-n binomial counts are bounded at n, whereas first-success waiting times have a long right tail when p is small.
Legacy practice clinic 22: expected-value language
Never write “the expected value will happen.” Say that the average count or waiting time approaches the expectation over many independent repetitions of the full process.
Legacy practice clinic 23: design before formula
If the problem statement does not establish independence or a stable p, note the missing assumption. Statistical modeling is conditional on a credible data-generating process, not on keyword matching.
Binomial and Geometric Practice Problems: Legacy Comparison: multiple-choice practice
Question 1. Binomial vs. Geometric Distribution
A random variable in a public health department in Central County during a follow-up evaluation period counts how many trials are required to obtain the first success, with constant success probability 0.205. Choose the model and justify it.
Answer: B
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.205.
Question 2. Binomial vs. Geometric Distribution
A random variable in a county election office in Capital Region during a community outreach cycle counts the number of successes among exactly 10 trials, with constant success probability 0.569. Choose the model and justify it.
Answer: D
The model is binomial. The number of trials is fixed at n=10, X counts successes, and the constant success probability is p=0.569.
Question 3. Binomial vs. Geometric Distribution
A random variable in a wildlife clinic in Midwest consortium during a weekday operations study counts how many trials are required to obtain the first success, with constant success probability 0.599. Choose the model and justify it.
Answer: A
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.599.
Question 4. Binomial vs. Geometric Distribution
A random variable in a farm cooperative in Pacific Northwest during a spring 2027 pilot counts how many trials are required to obtain the first success, with constant success probability 0.261. Choose the model and justify it.
Answer: C
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.261.
Question 5. Binomial vs. Geometric Distribution
A random variable in a municipal emergency dispatch center in North Valley during a follow-up evaluation period counts how many trials are required to obtain the first success, with constant success probability 0.407. Choose the model and justify it.
Answer: B
Binomial and Geometric Practice Problems: Legacy Comparison — Question 5. Binomial vs. Geometric Distribution: The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.407.
Question 6. Binomial vs. Geometric Distribution
A random variable in a solar installer in Pine Ridge during a randomized pilot period counts how many trials are required to obtain the first success, with constant success probability 0.565. Choose the model and justify it.
Answer: D
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.565.
Question 7. Binomial vs. Geometric Distribution
A random variable in a grocery cooperative in Sunbelt district during a quarterly performance study counts how many trials are required to obtain the first success, with constant success probability 0.475. Choose the model and justify it.
Answer: B
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.475.
Question 8. Binomial vs. Geometric Distribution
A random variable in a community college in Pine Ridge during a semester-long cohort study counts the number of successes among exactly 16 trials, with constant success probability 0.358. Choose the model and justify it.
Answer: A
The model is binomial. The number of trials is fixed at n=16, X counts successes, and the constant success probability is p=0.358.
Question 9. Binomial vs. Geometric Distribution
A random variable in a county library in Westview during a six-week field trial counts the number of successes among exactly 11 trials, with constant success probability 0.258. Choose the model and justify it.
Answer: A
The model is binomial. The number of trials is fixed at n=11, X counts successes, and the constant success probability is p=0.258.
Question 10. Binomial vs. Geometric Distribution
A random variable in a regional airport authority in Cedar Grove during a quarterly performance study counts the number of successes among exactly 26 trials, with constant success probability 0.332. Choose the model and justify it.
Answer: B
The model is binomial. The number of trials is fixed at n=26, X counts successes, and the constant success probability is p=0.332.
Question 11. Binomial vs. Geometric Distribution
A random variable in a university advising center in Riverbend during a community outreach cycle counts the number of successes among exactly 12 trials, with constant success probability 0.344. Choose the model and justify it.
Answer: B
The model is binomial. The number of trials is fixed at n=12, X counts successes, and the constant success probability is p=0.344.
Question 12. Binomial vs. Geometric Distribution
A random variable in a regional hospital in Desert County during a multiweek validation study counts how many trials are required to obtain the first success, with constant success probability 0.587. Choose the model and justify it.
Answer: C
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.587.
Question 13. Binomial vs. Geometric Distribution
A random variable in a housing authority in Pine Ridge during a yearly program evaluation counts how many trials are required to obtain the first success, with constant success probability 0.239. Choose the model and justify it.
Answer: B
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.239.
Question 14. Binomial vs. Geometric Distribution
A random variable in a county library in South Harbor during a community outreach cycle counts the number of successes among exactly 18 trials, with constant success probability 0.304. Choose the model and justify it.
Answer: D
The model is binomial. The number of trials is fixed at n=18, X counts successes, and the constant success probability is p=0.304.
Question 15. Binomial vs. Geometric Distribution
A random variable in a city transit agency in Atlantic Corridor during a weekday operations study counts how many trials are required to obtain the first success, with constant success probability 0.399. Choose the model and justify it.
Answer: A
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.399.
Question 16. Binomial vs. Geometric Distribution
A random variable in a solar installer in Atlantic Corridor during a multiweek validation study counts the number of successes among exactly 25 trials, with constant success probability 0.511. Choose the model and justify it.
Answer: C
The model is binomial. The number of trials is fixed at n=25, X counts successes, and the constant success probability is p=0.511.
Question 17. Binomial vs. Geometric Distribution
A random variable in a regional manufacturer in Metro East during a weekday operations study counts the number of successes among exactly 25 trials, with constant success probability 0.383. Choose the model and justify it.
Answer: B
The model is binomial. The number of trials is fixed at n=25, X counts successes, and the constant success probability is p=0.383.
Question 18. Binomial vs. Geometric Distribution
A random variable in a city recreation department in South Harbor during a spring 2027 pilot counts how many trials are required to obtain the first success, with constant success probability 0.303. Choose the model and justify it.
Answer: B
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.303.
Question 19. Binomial vs. Geometric Distribution
A random variable in a community bank in Capital Region during a summer implementation review counts how many trials are required to obtain the first success, with constant success probability 0.409. Choose the model and justify it.
Answer: B
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.409.
Question 20. Binomial vs. Geometric Distribution
A random variable in a state park in Capital Region during a randomized pilot period counts how many trials are required to obtain the first success, with constant success probability 0.365. Choose the model and justify it.
Answer: D
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.365.
Question 21. Binomial vs. Geometric Distribution
A random variable in a university advising center in Great Lakes during a two-month observation window counts the number of successes among exactly 19 trials, with constant success probability 0.634. Choose the model and justify it.
Answer: D
The model is binomial. The number of trials is fixed at n=19, X counts successes, and the constant success probability is p=0.634.
Question 22. Binomial vs. Geometric Distribution
A random variable in a grocery cooperative in Lakeside district during a randomized pilot period counts the number of successes among exactly 14 trials, with constant success probability 0.21. Choose the model and justify it.
Answer: B
The model is binomial. The number of trials is fixed at n=14, X counts successes, and the constant success probability is p=0.21.
Question 23. Binomial vs. Geometric Distribution
A random variable in a recycling program in Prairie District during a weekday operations study counts the number of successes among exactly 20 trials, with constant success probability 0.527. Choose the model and justify it.
Answer: A
The model is binomial. The number of trials is fixed at n=20, X counts successes, and the constant success probability is p=0.527.
Question 24. Binomial vs. Geometric Distribution
A random variable in a regional manufacturer in Mountain Region during a regional benchmarking study counts the number of successes among exactly 13 trials, with constant success probability 0.514. Choose the model and justify it.
Answer: B
The model is binomial. The number of trials is fixed at n=13, X counts successes, and the constant success probability is p=0.514.
Question 25. Binomial vs. Geometric Distribution
A random variable in a county library in Westview during a service-improvement study counts the number of successes among exactly 21 trials, with constant success probability 0.359. Choose the model and justify it.
Answer: C
The model is binomial. The number of trials is fixed at n=21, X counts successes, and the constant success probability is p=0.359.
Question 26. Binomial vs. Geometric Distribution
A random variable in a regional hospital in Westview during a multiweek validation study counts the number of successes among exactly 18 trials, with constant success probability 0.363. Choose the model and justify it.
Answer: D
The model is binomial. The number of trials is fixed at n=18, X counts successes, and the constant success probability is p=0.363.
Question 27. Binomial vs. Geometric Distribution
A random variable in a regional hospital in Prairie District during a community outreach cycle counts the number of successes among exactly 20 trials, with constant success probability 0.462. Choose the model and justify it.
Answer: D
The model is binomial. The number of trials is fixed at n=20, X counts successes, and the constant success probability is p=0.462.
Question 28. Binomial vs. Geometric Distribution
A random variable in a community bank in Pacific Northwest during a yearly program evaluation counts the number of successes among exactly 20 trials, with constant success probability 0.451. Choose the model and justify it.
Answer: C
The model is binomial. The number of trials is fixed at n=20, X counts successes, and the constant success probability is p=0.451.
Question 29. Binomial vs. Geometric Distribution
A random variable in a community college in Great Lakes during a randomized pilot period counts the number of successes among exactly 22 trials, with constant success probability 0.539. Choose the model and justify it.
Answer: D
The model is binomial. The number of trials is fixed at n=22, X counts successes, and the constant success probability is p=0.539.
Question 30. Binomial vs. Geometric Distribution
A random variable in a state park in Cedar Grove during a follow-up evaluation period counts how many trials are required to obtain the first success, with constant success probability 0.343. Choose the model and justify it.
Answer: A
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.343.
Question 31. Binomial vs. Geometric Distribution
A random variable in a public high school in North Valley during a monthly quality review counts how many trials are required to obtain the first success, with constant success probability 0.315. Choose the model and justify it.
Answer: C
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.315.
Question 32. Binomial vs. Geometric Distribution
A random variable in a public health department in Pacific Northwest during a regional benchmarking study counts the number of successes among exactly 27 trials, with constant success probability 0.204. Choose the model and justify it.
Answer: D
The model is binomial. The number of trials is fixed at n=27, X counts successes, and the constant success probability is p=0.204.
Question 33. Binomial vs. Geometric Distribution
A random variable in a community bank in Atlantic Corridor during a semester-long cohort study counts how many trials are required to obtain the first success, with constant success probability 0.349. Choose the model and justify it.
Answer: C
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.349.
Question 34. Binomial vs. Geometric Distribution
A random variable in a municipal water office in Lakeside district during a multiweek validation study counts how many trials are required to obtain the first success, with constant success probability 0.49. Choose the model and justify it.
Answer: A
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.49.
Question 35. Binomial vs. Geometric Distribution
A random variable in a county library in Pine Ridge during a community outreach cycle counts the number of successes among exactly 10 trials, with constant success probability 0.251. Choose the model and justify it.
Answer: A
The model is binomial. The number of trials is fixed at n=10, X counts successes, and the constant success probability is p=0.251.
Question 36. Binomial vs. Geometric Distribution
A random variable in a city recreation department in Pine Ridge during a six-week field trial counts the number of successes among exactly 27 trials, with constant success probability 0.205. Choose the model and justify it.
Answer: C
The model is binomial. The number of trials is fixed at n=27, X counts successes, and the constant success probability is p=0.205.
Binomial and Geometric Practice Problems: Legacy Comparison: free-response practice
FRQ set 1: Binomial vs. Geometric Distribution
Scenario. A random variable in a regional airport authority in Desert County during a pre-exam training cycle counts the number of successes among exactly 20 trials, with constant success probability 0.379. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is binomial. The number of trials is fixed at n=20, X counts successes, and the constant success probability is p=0.379.
FRQ set 2: Binomial vs. Geometric Distribution
Scenario. A random variable in a community bank in New England network during a monthly quality review counts how many trials are required to obtain the first success, with constant success probability 0.438. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.438.
FRQ set 3: Binomial vs. Geometric Distribution
Scenario. A random variable in a municipal water office in Westview during a weekday operations study counts the number of successes among exactly 14 trials, with constant success probability 0.524. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is binomial. The number of trials is fixed at n=14, X counts successes, and the constant success probability is p=0.524.
FRQ set 4: Binomial vs. Geometric Distribution
Scenario. A random variable in a university advising center in Midwest consortium during a school-year data collection counts the number of successes among exactly 30 trials, with constant success probability 0.233. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is binomial. The number of trials is fixed at n=30, X counts successes, and the constant success probability is p=0.233.
FRQ set 5: Binomial vs. Geometric Distribution
Scenario. A random variable in a digital learning platform in North Valley during a regional benchmarking study counts how many trials are required to obtain the first success, with constant success probability 0.369. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.369.
FRQ set 6: Binomial vs. Geometric Distribution
Scenario. A random variable in a public high school in Westview during a follow-up evaluation period counts the number of successes among exactly 26 trials, with constant success probability 0.422. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is binomial. The number of trials is fixed at n=26, X counts successes, and the constant success probability is p=0.422.
FRQ set 7: Binomial vs. Geometric Distribution
Scenario. A random variable in a university advising center in Westview during a pre-exam training cycle counts how many trials are required to obtain the first success, with constant success probability 0.623. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.623.
FRQ set 8: Binomial vs. Geometric Distribution
Scenario. A random variable in a regional airport authority in Riverbend during a spring 2027 pilot counts how many trials are required to obtain the first success, with constant success probability 0.579. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.579.
FRQ set 9: Binomial vs. Geometric Distribution
Scenario. A random variable in a farm cooperative in Central County during a randomized pilot period counts how many trials are required to obtain the first success, with constant success probability 0.202. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.202.
FRQ set 10: Binomial vs. Geometric Distribution
Scenario. A random variable in a city transit agency in Westview during a baseline measurement week counts how many trials are required to obtain the first success, with constant success probability 0.588. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.588.
FRQ set 11: Binomial vs. Geometric Distribution
Scenario. A random variable in a county library in Prairie District during a fall 2026 audit counts how many trials are required to obtain the first success, with constant success probability 0.504. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.504.
FRQ set 12: Binomial vs. Geometric Distribution
Scenario. A random variable in a regional manufacturer in Pine Ridge during a quarterly performance study counts the number of successes among exactly 16 trials, with constant success probability 0.382. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is binomial. The number of trials is fixed at n=16, X counts successes, and the constant success probability is p=0.382.
FRQ set 13: Binomial vs. Geometric Distribution
Scenario. A random variable in a public health department in Atlantic Corridor during a semester-long cohort study counts the number of successes among exactly 10 trials, with constant success probability 0.64. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is binomial. The number of trials is fixed at n=10, X counts successes, and the constant success probability is p=0.64.
FRQ set 14: Binomial vs. Geometric Distribution
Scenario. A random variable in a recycling program in Coastal Plains during a two-month observation window counts how many trials are required to obtain the first success, with constant success probability 0.275. Choose the model and justify it.
- Define the event or random variable and state the requested probability or long-run quantity.
- Verify the relevant model conditions or probability-distribution requirements.
- Show the probability, expected-value, or distribution calculation with labeled terms.
- Interpret the result as a probability or long-run behavior and distinguish it from a guarantee for one trial.
Model response
The model is geometric. The stopping time is random, X counts trials until the first success, and the constant success probability is p=0.275.
