AP®StatisticsCollege BoardExam QuestionsUnit 4: Probability, Random Variables & Probability DistributionsBinomial & Geometric Distributions

Binomial & Geometric Distributions (College Board AP® Statistics)

Exam Questions

5 mins5 questions
1
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Fifty arrows are fired at a target. Each arrow is fired from the same spot, and the same type of bow and arrows are used each time, so whether the arrow hits the target is independent from one attempt to the next. The arrow hitting the target is solely based on the ability of the person firing it. Let X represent the number of arrows that hit the target out of the 50 arrows fired. Under which of the following situations would X be a binomial random variable?

  • All arrows are fired by the same person, whose ability improved as they fired more arrows.

  • Fifty people of different abilities each fired one arrow.

  • Five people of equal ability each fired ten arrows, and their abilities did not change.

  • Two people of different abilities each fired 25 arrows.

  • Two people of the same ability fired arrows at the target. One person fired arrows until they missed the target, and then the other person did the same. This continued until 50 arrows were fired.

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2
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A teacher creates a quiz consisting of 20 multiple-choice questions. Each question has five options. A student passes the quiz if they get at least 60 percent of the questions correct. The teacher wants to reduce the chance of a student passing by randomly guessing the answer to all of the questions. Which of the following options would help the teacher achieve this?

  • Change the rule so that students pass if they answer less than 40 percent of the questions incorrectly.

  • Decrease the number of options per question.

  • Decrease the percentage of questions that a student must get correct in order to pass.

  • Increase the number of questions in the quiz.

  • Change the order of the questions in the quiz.

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3
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Which of the following graphs represents a geometric distribution with p equals 0.5?

  • Bar chart showing the probability distribution for the number of successes (1-10), with each probability value being 0.25.
  • Bar graph showing probability distribution with number of successes (1–10) on the x-axis and probability (0 to 0.25) on the y-axis, peaking at one success.
  • Bar chart showing the probabilities of achieving 0-10 successes. The highest probability is around 0.75 for one success, decreasing sharply for more successes.
  • Bar graph showing the probability of the number of successes. Probability decreases from about 0.25 at 1 success to below 0.05 at 10 successes.
  • Bar graph showing the probability distribution of the number of successes (1-10). The probability decreases as the number of successes increases. The height of the first bar is 0.75.

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4
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In a game show, there are six contestants and three boxes. One box contains a gold token, and the other two are empty. All contestants must choose a box independent of each other. Once all of the contestants have made their choices, the boxes are opened, and the contestants who chose the box with the gold token progress to the next round of the show.

What is the probability that at least five of the six contestants will progress to the next round?

  • 0.0014

  • 0.0165

  • 0.0178

  • 0.9822

  • 0.9986

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5
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Mr. White estimates that 30 percent of students at his school know how to play chess. Mr. White randomly selects ten students. Assuming that Mr. White's estimate is correct, what is the probability that at least two students in the sample know how to play chess?

  • 0.1493

  • 0.2335

  • 0.3828

  • 0.6172

  • 0.8507

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