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Binomial and Geometric Distribution Practice Problems with Solutions

Learn binomial and geometric distribution practice problems with current AP Statistics scope, proper formulas, worked examples, and original Easy, Tough, a.

Published July 18, 2026Updated July 18, 2026By Engr. Muhammad Yar Saqib43 min read

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Practice Test

Binomial and Geometric Distribution Practice Problems with Solutions

An original practice environment for mixed binomial and geometric practice, with clear instructions, answer access, scoring evidence, and diagnostic review.

Course status: Legacy enrichment
Updated: July 18, 2026
Practice: Easy, Tough and Toughest

Practice Set Instructions: Binomial And Geometric Distribution Practice Problems

Mixed binomial-geometric practice must begin by identifying the random variable and checking the trial structure, with every geometric item labeled outside the revised May 2027 core.

Reader taskmodel identification, hand calculation, calculator work, and complete solutions

Planned modules7

Mathematics2 expressions

Worked checks60

Boundary: Keep every geometric item marked as legacy for May 2027.

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Questions, Answers and Diagnostic Evidence

Every question in Binomial and Geometric Distribution Practice Problems with Solutions is newly written from the revised framework and the logic visible in public College Board materials. Constructed numerical settings are identified as instructional scenarios and are never represented as measurements from a real population. No released or secure question wording is reproduced.

Easy Practice

Easy 1: Mixed AP-style MCQs

Question P48-Easy-1. For a water-filtration experiment, compare a fixed-14-trial count with a first-success-on-trial-3 variable when $p=0.52$ .

Worked solution and validity check

Worked solution P48-Easy-1. The fixed-trial count is binomial with probability 0.01595; the first-success variable is geometric with probability 0.11981. $\binom{14}{3}p^3(1-p)^{11}=0.01595,\qquad (1-p)^{2}p=0.11981.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 2: FRQs

Question P48-Easy-2. For a water-filtration experiment, compare a fixed-12-trial count with a first-success-on-trial-4 variable when $p=0.45$ .

Worked solution and validity check

Worked solution P48-Easy-2. The fixed-trial count is binomial with probability 0.16996; the first-success variable is geometric with probability 0.07487. $\binom{12}{4}p^4(1-p)^{8}=0.16996,\qquad (1-p)^{3}p=0.07487.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 3: Complete step-by-step solutions

Question P48-Easy-3. For a commuter route study, compare a fixed-16-trial count with a first-success-on-trial-5 variable when $p=0.44$ .

Worked solution and validity check

Worked solution P48-Easy-3. The fixed-trial count is binomial with probability 0.12235; the first-success variable is geometric with probability 0.04327. $\binom{16}{5}p^5(1-p)^{11}=0.12235,\qquad (1-p)^{4}p=0.04327.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 4: Question bank by difficulty

Question P48-Easy-4. For a city bus arrival investigation, compare a fixed-13-trial count with a first-success-on-trial-6 variable when $p=0.46$ .

Worked solution and validity check

Worked solution P48-Easy-4. The fixed-trial count is binomial with probability 0.21768; the first-success variable is geometric with probability 0.02112. $\binom{13}{6}p^6(1-p)^{7}=0.21768,\qquad (1-p)^{5}p=0.02112.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 5: Identify distribution

Question P48-Easy-5. For a public-parks visitor survey, compare a fixed-12-trial count with a first-success-on-trial-3 variable when $p=0.35$ .

Worked solution and validity check

Worked solution P48-Easy-5. The fixed-trial count is binomial with probability 0.19537; the first-success variable is geometric with probability 0.14788. $\binom{12}{3}p^3(1-p)^{9}=0.19537,\qquad (1-p)^{2}p=0.14788.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 6: Hand calculations

Question P48-Easy-6. For a classroom memory study, compare a fixed-16-trial count with a first-success-on-trial-4 variable when $p=0.54$ .

Worked solution and validity check

Worked solution P48-Easy-6. The fixed-trial count is binomial with probability 0.01389; the first-success variable is geometric with probability 0.05256. $\binom{16}{4}p^4(1-p)^{12}=0.01389,\qquad (1-p)^{3}p=0.05256.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 7: Calculator calculations

Question P48-Easy-7. For a city bus arrival investigation, compare a fixed-13-trial count with a first-success-on-trial-5 variable when $p=0.41$ .

Worked solution and validity check

Worked solution P48-Easy-7. The fixed-trial count is binomial with probability 0.21893; the first-success variable is geometric with probability 0.04968. $\binom{13}{5}p^5(1-p)^{8}=0.21893,\qquad (1-p)^{4}p=0.04968.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 8: Mixed AP-style MCQs

Question P48-Easy-8. For a water-filtration experiment, compare a fixed-12-trial count with a first-success-on-trial-6 variable when $p=0.45$ .

Worked solution and validity check

Worked solution P48-Easy-8. The fixed-trial count is binomial with probability 0.21238; the first-success variable is geometric with probability 0.02265. $\binom{12}{6}p^6(1-p)^{6}=0.21238,\qquad (1-p)^{5}p=0.02265.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 9: FRQs

Question P48-Easy-9. For a website response-time study, compare a fixed-13-trial count with a first-success-on-trial-3 variable when $p=0.46$ .

Worked solution and validity check

Worked solution P48-Easy-9. The fixed-trial count is binomial with probability 0.05869; the first-success variable is geometric with probability 0.13414. $\binom{13}{3}p^3(1-p)^{10}=0.05869,\qquad (1-p)^{2}p=0.13414.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 10: Complete step-by-step solutions

Question P48-Easy-10. For a package-delivery sample, compare a fixed-16-trial count with a first-success-on-trial-4 variable when $p=0.44$ .

Worked solution and validity check

Worked solution P48-Easy-10. The fixed-trial count is binomial with probability 0.06488; the first-success variable is geometric with probability 0.07727. $\binom{16}{4}p^4(1-p)^{12}=0.06488,\qquad (1-p)^{3}p=0.07727.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 11: Question bank by difficulty

Question P48-Easy-11. For a classroom memory study, compare a fixed-13-trial count with a first-success-on-trial-5 variable when $p=0.51$ .

Worked solution and validity check

Worked solution P48-Easy-11. The fixed-trial count is binomial with probability 0.14757; the first-success variable is geometric with probability 0.02940. $\binom{13}{5}p^5(1-p)^{8}=0.14757,\qquad (1-p)^{4}p=0.02940.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 12: Identify distribution

Question P48-Easy-12. For a greenhouse germination experiment, compare a fixed-16-trial count with a first-success-on-trial-6 variable when $p=0.39$ .

Worked solution and validity check

Worked solution P48-Easy-12. The fixed-trial count is binomial with probability 0.20101; the first-success variable is geometric with probability 0.03294. $\binom{16}{6}p^6(1-p)^{10}=0.20101,\qquad (1-p)^{5}p=0.03294.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 13: Hand calculations

Question P48-Easy-13. For a classroom memory study, compare a fixed-16-trial count with a first-success-on-trial-3 variable when $p=0.39$ .

Worked solution and validity check

Worked solution P48-Easy-13. The fixed-trial count is binomial with probability 0.05379; the first-success variable is geometric with probability 0.14512. $\binom{16}{3}p^3(1-p)^{13}=0.05379,\qquad (1-p)^{2}p=0.14512.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 14: Calculator calculations

Question P48-Easy-14. For a quality-control inspection, compare a fixed-12-trial count with a first-success-on-trial-4 variable when $p=0.45$ .

Worked solution and validity check

Worked solution P48-Easy-14. The fixed-trial count is binomial with probability 0.16996; the first-success variable is geometric with probability 0.07487. $\binom{12}{4}p^4(1-p)^{8}=0.16996,\qquad (1-p)^{3}p=0.07487.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 15: Mixed AP-style MCQs

Question P48-Easy-15. For a public-parks visitor survey, compare a fixed-14-trial count with a first-success-on-trial-5 variable when $p=0.37$ .

Worked solution and validity check

Worked solution P48-Easy-15. The fixed-trial count is binomial with probability 0.21704; the first-success variable is geometric with probability 0.05829. $\binom{14}{5}p^5(1-p)^{9}=0.21704,\qquad (1-p)^{4}p=0.05829.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 16: FRQs

Question P48-Easy-16. For a commuter route study, compare a fixed-13-trial count with a first-success-on-trial-6 variable when $p=0.51$ .

Worked solution and validity check

Worked solution P48-Easy-16. The fixed-trial count is binomial with probability 0.20479; the first-success variable is geometric with probability 0.01441. $\binom{13}{6}p^6(1-p)^{7}=0.20479,\qquad (1-p)^{5}p=0.01441.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 17: Complete step-by-step solutions

Question P48-Easy-17. For a commuter route study, compare a fixed-16-trial count with a first-success-on-trial-3 variable when $p=0.54$ .

Worked solution and validity check

Worked solution P48-Easy-17. The fixed-trial count is binomial with probability 0.00364; the first-success variable is geometric with probability 0.11426. $\binom{16}{3}p^3(1-p)^{13}=0.00364,\qquad (1-p)^{2}p=0.11426.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 18: Question bank by difficulty

Question P48-Easy-18. For a classroom memory study, compare a fixed-14-trial count with a first-success-on-trial-4 variable when $p=0.42$ .

Worked solution and validity check

Worked solution P48-Easy-18. The fixed-trial count is binomial with probability 0.13419; the first-success variable is geometric with probability 0.08195. $\binom{14}{4}p^4(1-p)^{10}=0.13419,\qquad (1-p)^{3}p=0.08195.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 19: Identify distribution

Question P48-Easy-19. For a greenhouse germination experiment, compare a fixed-13-trial count with a first-success-on-trial-5 variable when $p=0.46$ .

Worked solution and validity check

Worked solution P48-Easy-19. The fixed-trial count is binomial with probability 0.19165; the first-success variable is geometric with probability 0.03911. $\binom{13}{5}p^5(1-p)^{8}=0.19165,\qquad (1-p)^{4}p=0.03911.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Easy 20: Hand calculations

Question P48-Easy-20. For a campus dining survey, compare a fixed-14-trial count with a first-success-on-trial-6 variable when $p=0.52$ .

Worked solution and validity check

Worked solution P48-Easy-20. The fixed-trial count is binomial with probability 0.16730; the first-success variable is geometric with probability 0.01325. $\binom{14}{6}p^6(1-p)^{8}=0.16730,\qquad (1-p)^{5}p=0.01325.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough Practice

Tough 1: Complete step-by-step solutions

Question P48-Tough-1. For a seedling-growth comparison, compare a fixed-12-trial count with a first-success-on-trial-3 variable when $p=0.45$ .

Worked solution and validity check

Worked solution P48-Tough-1. The fixed-trial count is binomial with probability 0.09233; the first-success variable is geometric with probability 0.13612. $\binom{12}{3}p^3(1-p)^{9}=0.09233,\qquad (1-p)^{2}p=0.13612.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 2: Question bank by difficulty

Question P48-Tough-2. For a package-delivery sample, compare a fixed-16-trial count with a first-success-on-trial-4 variable when $p=0.54$ .

Worked solution and validity check

Worked solution P48-Tough-2. The fixed-trial count is binomial with probability 0.01389; the first-success variable is geometric with probability 0.05256. $\binom{16}{4}p^4(1-p)^{12}=0.01389,\qquad (1-p)^{3}p=0.05256.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 3: Identify distribution

Question P48-Tough-3. For a campus dining survey, compare a fixed-12-trial count with a first-success-on-trial-5 variable when $p=0.45$ .

Worked solution and validity check

Worked solution P48-Tough-3. The fixed-trial count is binomial with probability 0.22250; the first-success variable is geometric with probability 0.04118. $\binom{12}{5}p^5(1-p)^{7}=0.22250,\qquad (1-p)^{4}p=0.04118.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 4: Hand calculations

Question P48-Tough-4. For a commuter route study, compare a fixed-15-trial count with a first-success-on-trial-6 variable when $p=0.43$ .

Worked solution and validity check

Worked solution P48-Tough-4. The fixed-trial count is binomial with probability 0.20095; the first-success variable is geometric with probability 0.02587. $\binom{15}{6}p^6(1-p)^{9}=0.20095,\qquad (1-p)^{5}p=0.02587.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 5: Calculator calculations

Question P48-Tough-5. For a reading-speed investigation, compare a fixed-16-trial count with a first-success-on-trial-3 variable when $p=0.54$ .

Worked solution and validity check

Worked solution P48-Tough-5. The fixed-trial count is binomial with probability 0.00364; the first-success variable is geometric with probability 0.11426. $\binom{16}{3}p^3(1-p)^{13}=0.00364,\qquad (1-p)^{2}p=0.11426.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 6: Mixed AP-style MCQs

Question P48-Tough-6. For a recycling-behavior survey, compare a fixed-14-trial count with a first-success-on-trial-4 variable when $p=0.37$ .

Worked solution and validity check

Worked solution P48-Tough-6. The fixed-trial count is binomial with probability 0.18478; the first-success variable is geometric with probability 0.09252. $\binom{14}{4}p^4(1-p)^{10}=0.18478,\qquad (1-p)^{3}p=0.09252.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 7: FRQs

Question P48-Tough-7. For a reading-speed investigation, compare a fixed-12-trial count with a first-success-on-trial-5 variable when $p=0.50$ .

Worked solution and validity check

Worked solution P48-Tough-7. The fixed-trial count is binomial with probability 0.19336; the first-success variable is geometric with probability 0.03125. $\binom{12}{5}p^5(1-p)^{7}=0.19336,\qquad (1-p)^{4}p=0.03125.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 8: Complete step-by-step solutions

Question P48-Tough-8. For a quality-control inspection, compare a fixed-14-trial count with a first-success-on-trial-6 variable when $p=0.37$ .

Worked solution and validity check

Worked solution P48-Tough-8. The fixed-trial count is binomial with probability 0.19120; the first-success variable is geometric with probability 0.03672. $\binom{14}{6}p^6(1-p)^{8}=0.19120,\qquad (1-p)^{5}p=0.03672.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 9: Question bank by difficulty

Question P48-Tough-9. For a school library checkout study, compare a fixed-16-trial count with a first-success-on-trial-3 variable when $p=0.39$ .

Worked solution and validity check

Worked solution P48-Tough-9. The fixed-trial count is binomial with probability 0.05379; the first-success variable is geometric with probability 0.14512. $\binom{16}{3}p^3(1-p)^{13}=0.05379,\qquad (1-p)^{2}p=0.14512.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 10: Identify distribution

Question P48-Tough-10. For a water-filtration experiment, compare a fixed-14-trial count with a first-success-on-trial-4 variable when $p=0.37$ .

Worked solution and validity check

Worked solution P48-Tough-10. The fixed-trial count is binomial with probability 0.18478; the first-success variable is geometric with probability 0.09252. $\binom{14}{4}p^4(1-p)^{10}=0.18478,\qquad (1-p)^{3}p=0.09252.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 11: Hand calculations

Question P48-Tough-11. For a reading-speed investigation, compare a fixed-12-trial count with a first-success-on-trial-5 variable when $p=0.50$ .

Worked solution and validity check

Worked solution P48-Tough-11. The fixed-trial count is binomial with probability 0.19336; the first-success variable is geometric with probability 0.03125. $\binom{12}{5}p^5(1-p)^{7}=0.19336,\qquad (1-p)^{4}p=0.03125.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 12: Calculator calculations

Question P48-Tough-12. For a package-delivery sample, compare a fixed-16-trial count with a first-success-on-trial-6 variable when $p=0.54$ .

Worked solution and validity check

Worked solution P48-Tough-12. The fixed-trial count is binomial with probability 0.08423; the first-success variable is geometric with probability 0.01112. $\binom{16}{6}p^6(1-p)^{10}=0.08423,\qquad (1-p)^{5}p=0.01112.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 13: Mixed AP-style MCQs

Question P48-Tough-13. For a reading-speed investigation, compare a fixed-12-trial count with a first-success-on-trial-3 variable when $p=0.40$ .

Worked solution and validity check

Worked solution P48-Tough-13. The fixed-trial count is binomial with probability 0.14189; the first-success variable is geometric with probability 0.14400. $\binom{12}{3}p^3(1-p)^{9}=0.14189,\qquad (1-p)^{2}p=0.14400.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 14: FRQs

Question P48-Tough-14. For a water-filtration experiment, compare a fixed-13-trial count with a first-success-on-trial-4 variable when $p=0.41$ .

Worked solution and validity check

Worked solution P48-Tough-14. The fixed-trial count is binomial with probability 0.17503; the first-success variable is geometric with probability 0.08421. $\binom{13}{4}p^4(1-p)^{9}=0.17503,\qquad (1-p)^{3}p=0.08421.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 15: Complete step-by-step solutions

Question P48-Tough-15. For a quality-control inspection, compare a fixed-12-trial count with a first-success-on-trial-5 variable when $p=0.40$ .

Worked solution and validity check

Worked solution P48-Tough-15. The fixed-trial count is binomial with probability 0.22703; the first-success variable is geometric with probability 0.05184. $\binom{12}{5}p^5(1-p)^{7}=0.22703,\qquad (1-p)^{4}p=0.05184.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 16: Question bank by difficulty

Question P48-Tough-16. For a greenhouse germination experiment, compare a fixed-16-trial count with a first-success-on-trial-6 variable when $p=0.49$ .

Worked solution and validity check

Worked solution P48-Tough-16. The fixed-trial count is binomial with probability 0.13195; the first-success variable is geometric with probability 0.01691. $\binom{16}{6}p^6(1-p)^{10}=0.13195,\qquad (1-p)^{5}p=0.01691.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 17: Identify distribution

Question P48-Tough-17. For a quality-control inspection, compare a fixed-13-trial count with a first-success-on-trial-3 variable when $p=0.46$ .

Worked solution and validity check

Worked solution P48-Tough-17. The fixed-trial count is binomial with probability 0.05869; the first-success variable is geometric with probability 0.13414. $\binom{13}{3}p^3(1-p)^{10}=0.05869,\qquad (1-p)^{2}p=0.13414.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 18: Hand calculations

Question P48-Tough-18. For a manufacturing fill-volume check, compare a fixed-15-trial count with a first-success-on-trial-4 variable when $p=0.53$ .

Worked solution and validity check

Worked solution P48-Tough-18. The fixed-trial count is binomial with probability 0.02663; the first-success variable is geometric with probability 0.05503. $\binom{15}{4}p^4(1-p)^{11}=0.02663,\qquad (1-p)^{3}p=0.05503.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 19: Calculator calculations

Question P48-Tough-19. For a water-filtration experiment, compare a fixed-16-trial count with a first-success-on-trial-5 variable when $p=0.54$ .

Worked solution and validity check

Worked solution P48-Tough-19. The fixed-trial count is binomial with probability 0.03914; the first-success variable is geometric with probability 0.02418. $\binom{16}{5}p^5(1-p)^{11}=0.03914,\qquad (1-p)^{4}p=0.02418.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Tough 20: Mixed AP-style MCQs

Question P48-Tough-20. For a water-filtration experiment, compare a fixed-13-trial count with a first-success-on-trial-6 variable when $p=0.41$ .

Worked solution and validity check

Worked solution P48-Tough-20. The fixed-trial count is binomial with probability 0.20285; the first-success variable is geometric with probability 0.02931. $\binom{13}{6}p^6(1-p)^{7}=0.20285,\qquad (1-p)^{5}p=0.02931.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest Practice

Toughest 1: Question bank by difficulty

Question P48-Toughest-1. For a seedling-growth comparison, compare a fixed-15-trial count with a first-success-on-trial-3 variable when $p=0.53$ .

Worked solution and validity check

Worked solution P48-Toughest-1. The fixed-trial count is binomial with probability 0.00787; the first-success variable is geometric with probability 0.11708. $\binom{15}{3}p^3(1-p)^{12}=0.00787,\qquad (1-p)^{2}p=0.11708.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 2: Identify distribution

Question P48-Toughest-2. For a website response-time study, compare a fixed-16-trial count with a first-success-on-trial-4 variable when $p=0.49$ .

Worked solution and validity check

Worked solution P48-Toughest-2. The fixed-trial count is binomial with probability 0.03249; the first-success variable is geometric with probability 0.06500. $\binom{16}{4}p^4(1-p)^{12}=0.03249,\qquad (1-p)^{3}p=0.06500.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 3: Hand calculations

Question P48-Toughest-3. For a reading-speed investigation, compare a fixed-15-trial count with a first-success-on-trial-5 variable when $p=0.38$ .

Worked solution and validity check

Worked solution P48-Toughest-3. The fixed-trial count is binomial with probability 0.19971; the first-success variable is geometric with probability 0.05615. $\binom{15}{5}p^5(1-p)^{10}=0.19971,\qquad (1-p)^{4}p=0.05615.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 4: Calculator calculations

Question P48-Toughest-4. For a tutoring-program evaluation, compare a fixed-16-trial count with a first-success-on-trial-6 variable when $p=0.44$ .

Worked solution and validity check

Worked solution P48-Toughest-4. The fixed-trial count is binomial with probability 0.17625; the first-success variable is geometric with probability 0.02423. $\binom{16}{6}p^6(1-p)^{10}=0.17625,\qquad (1-p)^{5}p=0.02423.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 5: Mixed AP-style MCQs

Question P48-Toughest-5. For a commuter route study, compare a fixed-13-trial count with a first-success-on-trial-3 variable when $p=0.46$ .

Worked solution and validity check

Worked solution P48-Toughest-5. The fixed-trial count is binomial with probability 0.05869; the first-success variable is geometric with probability 0.13414. $\binom{13}{3}p^3(1-p)^{10}=0.05869,\qquad (1-p)^{2}p=0.13414.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 6: FRQs

Question P48-Toughest-6. For a campus dining survey, compare a fixed-16-trial count with a first-success-on-trial-4 variable when $p=0.44$ .

Worked solution and validity check

Worked solution P48-Toughest-6. The fixed-trial count is binomial with probability 0.06488; the first-success variable is geometric with probability 0.07727. $\binom{16}{4}p^4(1-p)^{12}=0.06488,\qquad (1-p)^{3}p=0.07727.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 7: Complete step-by-step solutions

Question P48-Toughest-7. For a manufacturing fill-volume check, compare a fixed-15-trial count with a first-success-on-trial-5 variable when $p=0.38$ .

Worked solution and validity check

Worked solution P48-Toughest-7. The fixed-trial count is binomial with probability 0.19971; the first-success variable is geometric with probability 0.05615. $\binom{15}{5}p^5(1-p)^{10}=0.19971,\qquad (1-p)^{4}p=0.05615.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 8: Question bank by difficulty

Question P48-Toughest-8. For a website response-time study, compare a fixed-15-trial count with a first-success-on-trial-6 variable when $p=0.43$ .

Worked solution and validity check

Worked solution P48-Toughest-8. The fixed-trial count is binomial with probability 0.20095; the first-success variable is geometric with probability 0.02587. $\binom{15}{6}p^6(1-p)^{9}=0.20095,\qquad (1-p)^{5}p=0.02587.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 9: Identify distribution

Question P48-Toughest-9. For a website response-time study, compare a fixed-12-trial count with a first-success-on-trial-3 variable when $p=0.50$ .

Worked solution and validity check

Worked solution P48-Toughest-9. The fixed-trial count is binomial with probability 0.05371; the first-success variable is geometric with probability 0.12500. $\binom{12}{3}p^3(1-p)^{9}=0.05371,\qquad (1-p)^{2}p=0.12500.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 10: Hand calculations

Question P48-Toughest-10. For a package-delivery sample, compare a fixed-15-trial count with a first-success-on-trial-4 variable when $p=0.38$ .

Worked solution and validity check

Worked solution P48-Toughest-10. The fixed-trial count is binomial with probability 0.14811; the first-success variable is geometric with probability 0.09056. $\binom{15}{4}p^4(1-p)^{11}=0.14811,\qquad (1-p)^{3}p=0.09056.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 11: Calculator calculations

Question P48-Toughest-11. For a city bus arrival investigation, compare a fixed-12-trial count with a first-success-on-trial-5 variable when $p=0.35$ .

Worked solution and validity check

Worked solution P48-Toughest-11. The fixed-trial count is binomial with probability 0.20392; the first-success variable is geometric with probability 0.06248. $\binom{12}{5}p^5(1-p)^{7}=0.20392,\qquad (1-p)^{4}p=0.06248.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 12: Mixed AP-style MCQs

Question P48-Toughest-12. For a battery-life laboratory trial, compare a fixed-16-trial count with a first-success-on-trial-6 variable when $p=0.39$ .

Worked solution and validity check

Worked solution P48-Toughest-12. The fixed-trial count is binomial with probability 0.20101; the first-success variable is geometric with probability 0.03294. $\binom{16}{6}p^6(1-p)^{10}=0.20101,\qquad (1-p)^{5}p=0.03294.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 13: FRQs

Question P48-Toughest-13. For a quality-control inspection, compare a fixed-15-trial count with a first-success-on-trial-3 variable when $p=0.38$ .

Worked solution and validity check

Worked solution P48-Toughest-13. The fixed-trial count is binomial with probability 0.08055; the first-success variable is geometric with probability 0.14607. $\binom{15}{3}p^3(1-p)^{12}=0.08055,\qquad (1-p)^{2}p=0.14607.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 14: Complete step-by-step solutions

Question P48-Toughest-14. For a classroom memory study, compare a fixed-16-trial count with a first-success-on-trial-4 variable when $p=0.44$ .

Worked solution and validity check

Worked solution P48-Toughest-14. The fixed-trial count is binomial with probability 0.06488; the first-success variable is geometric with probability 0.07727. $\binom{16}{4}p^4(1-p)^{12}=0.06488,\qquad (1-p)^{3}p=0.07727.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 15: Question bank by difficulty

Question P48-Toughest-15. For a greenhouse germination experiment, compare a fixed-13-trial count with a first-success-on-trial-5 variable when $p=0.36$ .

Worked solution and validity check

Worked solution P48-Toughest-15. The fixed-trial count is binomial with probability 0.21904; the first-success variable is geometric with probability 0.06040. $\binom{13}{5}p^5(1-p)^{8}=0.21904,\qquad (1-p)^{4}p=0.06040.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 16: Identify distribution

Question P48-Toughest-16. For a package-delivery sample, compare a fixed-16-trial count with a first-success-on-trial-6 variable when $p=0.54$ .

Worked solution and validity check

Worked solution P48-Toughest-16. The fixed-trial count is binomial with probability 0.08423; the first-success variable is geometric with probability 0.01112. $\binom{16}{6}p^6(1-p)^{10}=0.08423,\qquad (1-p)^{5}p=0.01112.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 17: Hand calculations

Question P48-Toughest-17. For a water-filtration experiment, compare a fixed-14-trial count with a first-success-on-trial-3 variable when $p=0.52$ .

Worked solution and validity check

Worked solution P48-Toughest-17. The fixed-trial count is binomial with probability 0.01595; the first-success variable is geometric with probability 0.11981. $\binom{14}{3}p^3(1-p)^{11}=0.01595,\qquad (1-p)^{2}p=0.11981.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 18: Calculator calculations

Question P48-Toughest-18. For a campus dining survey, compare a fixed-14-trial count with a first-success-on-trial-4 variable when $p=0.42$ .

Worked solution and validity check

Worked solution P48-Toughest-18. The fixed-trial count is binomial with probability 0.13419; the first-success variable is geometric with probability 0.08195. $\binom{14}{4}p^4(1-p)^{10}=0.13419,\qquad (1-p)^{3}p=0.08195.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 19: Mixed AP-style MCQs

Question P48-Toughest-19. For a greenhouse germination experiment, compare a fixed-12-trial count with a first-success-on-trial-5 variable when $p=0.50$ .

Worked solution and validity check

Worked solution P48-Toughest-19. The fixed-trial count is binomial with probability 0.19336; the first-success variable is geometric with probability 0.03125. $\binom{12}{5}p^5(1-p)^{7}=0.19336,\qquad (1-p)^{4}p=0.03125.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Toughest 20: FRQs

Question P48-Toughest-20. For a website response-time study, compare a fixed-12-trial count with a first-success-on-trial-6 variable when $p=0.45$ .

Worked solution and validity check

Worked solution P48-Toughest-20. The fixed-trial count is binomial with probability 0.21238; the first-success variable is geometric with probability 0.02265. $\binom{12}{6}p^6(1-p)^{6}=0.21238,\qquad (1-p)^{5}p=0.02265.$ Interpretation: The random variable definition, not the word success, determines the distribution. Validity: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule. Error to reject: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Question bank by difficulty

Use the completed practice set to diagnose Question bank by difficulty within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.01040; the first-success variable is geometric with probability 0.12745.

Scoring evidence

For Question bank by difficulty in binomial and geometric distribution practice problems, The random variable definition, not the word success, determines the distribution. Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

Distractor pattern

For Question bank by difficulty in binomial and geometric distribution practice problems, reject this error: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Open a worked diagnostic for Question bank by difficulty

For a school library checkout study, compare a fixed-16-trial count with a first-success-on-trial-3 variable when $p=0.49$ .

\binom{16}{3}p^3(1-p)^{13}=0.01040,\qquad (1-p)^{2}p=0.12745.

Identify distribution

Use the completed practice set to diagnose Identify distribution within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.16996; the first-success variable is geometric with probability 0.07487.

Scoring evidence

For Identify distribution in binomial and geometric distribution practice problems, The random variable definition, not the word success, determines the distribution. Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

Distractor pattern

For Identify distribution in binomial and geometric distribution practice problems, reject this error: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Open a worked diagnostic for Identify distribution

For a battery-life laboratory trial, compare a fixed-12-trial count with a first-success-on-trial-4 variable when $p=0.45$ .

\binom{12}{4}p^4(1-p)^{8}=0.16996,\qquad (1-p)^{3}p=0.07487.

Hand calculations

Use the completed practice set to diagnose Hand calculations within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.15151; the first-success variable is geometric with probability 0.03709.

Scoring evidence

For Hand calculations in binomial and geometric distribution practice problems, The random variable definition, not the word success, determines the distribution. Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

Distractor pattern

For Hand calculations in binomial and geometric distribution practice problems, reject this error: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Open a worked diagnostic for Hand calculations

For a package-delivery sample, compare a fixed-14-trial count with a first-success-on-trial-5 variable when $p=0.47$ .

\binom{14}{5}p^5(1-p)^{9}=0.15151,\qquad (1-p)^{4}p=0.03709.

Calculator calculations

Use the completed practice set to diagnose Calculator calculations within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.08423; the first-success variable is geometric with probability 0.01112.

Scoring evidence

For Calculator calculations in binomial and geometric distribution practice problems, The random variable definition, not the word success, determines the distribution. Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

Distractor pattern

For Calculator calculations in binomial and geometric distribution practice problems, reject this error: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Open a worked diagnostic for Calculator calculations

For a city bus arrival investigation, compare a fixed-16-trial count with a first-success-on-trial-6 variable when $p=0.54$ .

\binom{16}{6}p^6(1-p)^{10}=0.08423,\qquad (1-p)^{5}p=0.01112.

Mixed AP-style MCQs

Use the completed practice set to diagnose Mixed AP-style MCQs within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.15384; the first-success variable is geometric with probability 0.14746.

Scoring evidence

For Mixed AP-style MCQs in binomial and geometric distribution practice problems, The random variable definition, not the word success, determines the distribution. Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

Distractor pattern

For Mixed AP-style MCQs in binomial and geometric distribution practice problems, reject this error: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Open a worked diagnostic for Mixed AP-style MCQs

For a commuter route study, compare a fixed-13-trial count with a first-success-on-trial-3 variable when $p=0.36$ .

\binom{13}{3}p^3(1-p)^{10}=0.15384,\qquad (1-p)^{2}p=0.14746.

FRQs

Use the completed practice set to diagnose FRQs within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.03249; the first-success variable is geometric with probability 0.06500.

Scoring evidence

For FRQs in binomial and geometric distribution practice problems, The random variable definition, not the word success, determines the distribution. Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

Distractor pattern

For FRQs in binomial and geometric distribution practice problems, reject this error: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Open a worked diagnostic for FRQs

For a manufacturing fill-volume check, compare a fixed-16-trial count with a first-success-on-trial-4 variable when $p=0.49$ .

\binom{16}{4}p^4(1-p)^{12}=0.03249,\qquad (1-p)^{3}p=0.06500.

Complete step-by-step solutions

Use the completed practice set to diagnose Complete step-by-step solutions within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.06606; the first-success variable is geometric with probability 0.02586.

Scoring evidence

For Complete step-by-step solutions in binomial and geometric distribution practice problems, The random variable definition, not the word success, determines the distribution. Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

Distractor pattern

For Complete step-by-step solutions in binomial and geometric distribution practice problems, reject this error: Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Open a worked diagnostic for Complete step-by-step solutions

For a classroom memory study, compare a fixed-15-trial count with a first-success-on-trial-5 variable when $p=0.53$ .

\binom{15}{5}p^5(1-p)^{10}=0.06606,\qquad (1-p)^{4}p=0.02586.

Formula Review After the Practice Set

Cumulative binomial probability

P(X\le k)=\sum_{j=0}^k\binom njp^j(1-p)^{n-j}

Cumulative binomial probability in Binomial And Geometric Distribution Practice Problems: Define the event or random variable first; complements, conditioning, and trial assumptions determine which probability expression applies.

Geometric right-tail probability

P(Y>k)=(1-p)^k

Geometric right-tail probability in Binomial And Geometric Distribution Practice Problems: Define the event or random variable first; complements, conditioning, and trial assumptions determine which probability expression applies.

AP Response and Publication Checklist

Audit point	Required evidence for binomial and geometric distribution practice problems
Scope	Keep every geometric item marked as legacy for May 2027.
Method or source	Mixed binomial-geometric practice must begin by identifying the random variable and checking the trial structure, with every geometric item labeled outside the revised May 2027 core.
Calculation	$\binom{15}{3}p^3(1-p)^{12}=0.01967,\qquad (1-p)^{2}p=0.12979.$
Interpretation	The random variable definition, not the word success, determines the distribution.
Validity	Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.
Correction	Geometric content is no longer revised-course core, so label the comparison as legacy enrichment.

Frequently Asked Questions

How does question bank by difficulty work in binomial and geometric distribution practice problems?

Answer for binomial and geometric distribution practice problems and Question bank by difficulty. The fixed-trial count is binomial with probability 0.06738; the first-success variable is geometric with probability 0.14129. The random variable definition, not the word success, determines the distribution. The required validity evidence is: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

How does identify distribution work in binomial and geometric distribution practice problems?

Answer for binomial and geometric distribution practice problems and Identify distribution. The fixed-trial count is binomial with probability 0.09630; the first-success variable is geometric with probability 0.07963. The random variable definition, not the word success, determines the distribution. The required validity evidence is: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

How does hand calculations work in binomial and geometric distribution practice problems?

Answer for binomial and geometric distribution practice problems and Hand calculations. The fixed-trial count is binomial with probability 0.06606; the first-success variable is geometric with probability 0.02586. The random variable definition, not the word success, determines the distribution. The required validity evidence is: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

How does calculator calculations work in binomial and geometric distribution practice problems?

Answer for binomial and geometric distribution practice problems and Calculator calculations. The fixed-trial count is binomial with probability 0.20479; the first-success variable is geometric with probability 0.01441. The random variable definition, not the word success, determines the distribution. The required validity evidence is: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

How does mixed ap-style mcqs work in binomial and geometric distribution practice problems?

Answer for binomial and geometric distribution practice problems and Mixed AP-style MCQs. The fixed-trial count is binomial with probability 0.03027; the first-success variable is geometric with probability 0.12245. The random variable definition, not the word success, determines the distribution. The required validity evidence is: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

How does frqs work in binomial and geometric distribution practice problems?

Answer for binomial and geometric distribution practice problems and FRQs. The fixed-trial count is binomial with probability 0.18478; the first-success variable is geometric with probability 0.09252. The random variable definition, not the word success, determines the distribution. The required validity evidence is: Both models need independent equal-p binary trials; binomial fixes n, whereas geometric fixes the stopping rule.

How does ap statistics binomial and geometric distributions test connect to Binomial And Geometric Distribution Practice Problems?

ap statistics binomial and geometric distributions test within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.14189; the first-success variable is geometric with probability 0.14400. The random variable definition, not the word success, determines the distribution. For Question bank by difficulty, the controlling scope is: Keep every geometric item marked as legacy for May 2027.

How does ap stats binomial and geometric distribution test connect to Binomial And Geometric Distribution Practice Problems?

ap stats binomial and geometric distribution test within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.02663; the first-success variable is geometric with probability 0.05503. The random variable definition, not the word success, determines the distribution. For Identify distribution, the controlling scope is: Keep every geometric item marked as legacy for May 2027.

How does binomial and geometric distributions worksheet connect to Binomial And Geometric Distribution Practice Problems?

binomial and geometric distributions worksheet within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.22703; the first-success variable is geometric with probability 0.05184. The random variable definition, not the word success, determines the distribution. For Hand calculations, the controlling scope is: Keep every geometric item marked as legacy for May 2027.

How does ap statistics binomial and geometric distributions worksheet connect to Binomial And Geometric Distribution Practice Problems?

ap statistics binomial and geometric distributions worksheet within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.16428; the first-success variable is geometric with probability 0.03865. The random variable definition, not the word success, determines the distribution. For Calculator calculations, the controlling scope is: Keep every geometric item marked as legacy for May 2027.

How does ap statistics binomial and geometric distribution test connect to Binomial And Geometric Distribution Practice Problems?

ap statistics binomial and geometric distribution test within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.05869; the first-success variable is geometric with probability 0.13414. The random variable definition, not the word success, determines the distribution. For Mixed AP-style MCQs, the controlling scope is: Keep every geometric item marked as legacy for May 2027.

How does ap statistics binomial and geometric distribution worksheet connect to Binomial And Geometric Distribution Practice Problems?

ap statistics binomial and geometric distribution worksheet within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.01389; the first-success variable is geometric with probability 0.05256. The random variable definition, not the word success, determines the distribution. For FRQs, the controlling scope is: Keep every geometric item marked as legacy for May 2027.

How does ap statistics binomial and geometric distributions quiz connect to Binomial And Geometric Distribution Practice Problems?

ap statistics binomial and geometric distributions quiz within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.19434; the first-success variable is geometric with probability 0.04753. The random variable definition, not the word success, determines the distribution. For Complete step-by-step solutions, the controlling scope is: Keep every geometric item marked as legacy for May 2027.

How does ap stats binomial and geometric distribution review connect to Binomial And Geometric Distribution Practice Problems?

ap stats binomial and geometric distribution review within binomial and geometric distribution practice problems. The fixed-trial count is binomial with probability 0.12810; the first-success variable is geometric with probability 0.04061. The random variable definition, not the word success, determines the distribution. For Question bank by difficulty, the controlling scope is: Keep every geometric item marked as legacy for May 2027.

Sources

Administrative and curricular statements in Binomial and Geometric Distribution Practice Problems with Solutions were checked on July 18, 2026. The linked College Board pages control any later policy change; all instructional datasets in original questions are explicitly constructed rather than attributed to a real study.

AP Statistics Course and Exam Description, effective Fall 2026

Binomial And Geometric Distribution Practice Problems Conclusion

Mixed binomial-geometric practice must begin by identifying the random variable and checking the trial structure, with every geometric item labeled outside the revised May 2027 core. Mastery of binomial and geometric distribution practice problems therefore requires the exact evidence, mathematics, interpretation, and scope developed in this guide, while preserving this boundary: Keep every geometric item marked as legacy for May 2027.

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