chapter 8

Chapter 8: The Binomial and Geometric Distributions

1. An airplane has a front and a rear door that are both opened to allow passengers to exit when the plane lands. The plane has 100 passengers seated. The number of passengers exiting through the front door should have A) a binomial distribution with mean 50. B) a binomial distribution with 100 trials but success probability not equal to 0.5. C) a normal distribution with a standard deviation of 5. D) none of the above.

Ans: D Section: 8.1 The Binomial Distributions

2. A small class has 10 students. Five of the students are male and five are female. I write

the name of each student on a 3-by-5 card. The cards are shuffled thoroughly and I choose one at random, observe the name of the student, and replace it in the set. The cards are thoroughly reshuffled and I again choose a card at random, observe the name, and replace it in the set. This is done a total of four times. Let X be the number of cards observed in these four trials with a name corresponding to a male student. The random variable X has which of the following probability distributions? A) the normal distribution with mean 2 and variance 1. B) the binomial distribution with parameters n = 4 and p = 0.5. C) the uniform distribution on 0, 1, 2, 3, 4. D) none of the above.

Ans: B Section: 8.1 The Binomial Distributions

3. For which of the following counts would a binomial probability model be reasonable?

A) the number of phone calls received in a one-hour period. B) the number of hearts in a hand of five cards dealt from a standard deck of 52 cards

that has been thoroughly shuffled. C) the number of 7's in a randomly selected set of five random digits from a table of

random digits. D) all of the above.

Ans: C Section: 8.1 The Binomial Distributions

4. A set of 10 cards consists of five red cards and five black cards. The cards are shuffled

thoroughly and I choose one at random, observe its color, and replace it in the set. The cards are thoroughly reshuffled and I again choose a card at random, observe its color, and replace it in the set. This is done a total of four times. Let X be the number of red cards observed in these four trials. The mean of X is

A) 4. B) 2. C) 1. D) 0.5. Ans: B Section: 8.1 The Binomial Distributions

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5. A set of 10 cards consists of five red cards and five black cards. The cards are shuffled thoroughly and I choose six of these at random. Let X be the number of red cards observed in the six chosen. The random variable X has which of the following probability distributions? A) the normal distribution with mean 3 and variance 1.22. B) the binomial distribution with parameters n = 6 and p = 0.5. C) the uniform distribution of 0, 1, 2, 3, 4, 5, 6. D) none of the above.


6. If X is B(n =9, p = 1/3), the mean µX of X is

A) 6. B) 3. C) 2. D) 1.414. Ans: B Section: 8.1 The Binomial Distributions

7. If X is B(n =9, p = 1/3), the standard deviation σX of X is

A) 6. B) 3. C) 2. D) 1.414. Ans: D Section: 8.1 The Binomial Distributions

8. In a certain game of chance, your chances of winning are 0.2. If you play the game five

times and outcomes are independent, the probability that you win at most once is A) 0.0819. B) 0.2. C) 0.4096. D) 0.7373. Ans: D Section: 8.1 The Binomial Distributions

9. In a certain game of chance, your chances of winning are 0.2. If you play the game five

times and outcomes are independent, the probability that you win all five times is A) 0.6723. B) 0.3277. C) 0.04. D) 0.00032. Ans: D Section: 8.1 The Binomial Distributions

10. In a certain game of chance, your chances of winning are 0.2. You play the game five

times and outcomes are independent. Suppose it costs $1 to play the game. If you win, you receive $4 (for a net gain of $3). If you lose, you receive nothing (for a net loss of $1). Your expected winnings are

A) $3. B) $0. C) –$1. D) –$2. Ans: C Section: 8.1 The Binomial Distributions

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Use the following to answer questions 11-13: A survey asks a random sample of 1500 adults in Ohio if they support an increase in the state sales tax from 5% to 6%, with the additional revenue going to education. Let X denote the number in the sample that say they support the increase. Suppose that 40% of all adults in Ohio support the increase.

11. The mean of X is A) 5%. B) 0.40. C) 40. D) 600. Ans: D Section: 8.1 The Binomial Distributions

12. The standard deviation of X is

A) 360. B) 40. C) 24.49. D) 18.97. Ans: D Section: 8.1 The Binomial Distributions

13. The probability that X is more than 750 is

A) less than 0.0001. B) about 0.1. C) 0.4602. D) 0.50. Ans: A Section: 8.1 The Binomial Distributions

14. A fair coin (one for which both the probability of heads and the probability of tails are

0.5) is tossed six times. The probability that less than 1/3 of the tosses are heads is A) 0.33. B) 0.109. C) 0.09. D) 0.0043. Ans: B Section: 8.1 The Binomial Distributions

15. A fair coin (one for which both the probability of heads and the probability of tails are

0.5) is tossed 60 times. The probability that less than 1/3 of the tosses are heads is A) 0.33. B) 0.109. C) 0.09. D) 0.005. Ans: D Section: 8.1 The Binomial Distributions

16. Suppose we select an SRS of size n = 100 from a large population having proportion p

of successes. Let X be the number of successes in the sample. For which value of p would it be safe to assume the distribution of X is approximately normal?

A) 0.01. B) 1/9. C) 0.975. D) 0.9999. Ans: B Section: 8.1 The Binomial Distributions

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17. In a test of ESP (extrasensory perception), the experimenter looks at cards that are hidden from the subject. Each card contains either a star, a circle, a wavy line, or a square. An experimenter looks at each of 100 cards in turn, and the subject tries to read the experimenter's mind and name the shape on each. What is the probability that the subject gets more than 30 correct if the subject does not have ESP and is just guessing?

A) 0.3100. B) 0.2500. C) 0.1251. D) less than 0.0001. Ans: C Section: 8.1 The Binomial Distributions

18. A multiple-choice exam has 100 questions, each with five possible answers. If a student

is just guessing at all the answers, the probability that he or she gets more than 30 correct is

A) 0.3100. B) 0.2000. C) 0.1020. D) 0.0062. Ans: D Section: 8.1 The Binomial Distributions

19. Suppose X is a random variable with the B(n = 9, p = 1/3) distribution. The probability X

is either a 0 or a 1 is A) 0.6667. B) 0.3333. C) 0.1431. D) 0.1111. Ans: C Section: 8.1 The Binomial Distributions

20. Suppose X is a random variable with the B(n = 6, p = 2/3) distribution. The probability X

is at least 5 is A) 0.6667. B) 0.4444. C) 0.3512. D) 0.0878. Ans: C Section: 8.1 The Binomial Distributions

Use the following to answer questions 21-22: There are 20 multiple-choice questions on an exam, each having responses a, b, c, or d. Suppose a student guesses the answer to each question, and the guesses from question to question are independent. Let X be the number of questions for which the student has the same answer as the person sitting next to him on his right.

21. The distribution of X is A) B(20, .2). B) B(20, .25). C) B(4, .25). D) impossible to determine unless the student sitting next to him is also guessing.

Ans: B Section: 8.1 The Binomial Distributions

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22. The probability that X is zero is closest to A) 0.0032. B) 0.0243. C) 0.2373. D) 0.3277. Ans: A Section: 8.1 The Binomial Distributions

23. A college basketball player makes 80% of his free throws. At the end of a game, his

team is losing by two points. He is fouled attempting a three-point shot and is awarded three free throws. Assuming each free throw is independent, what is the probability that he makes at least two of the free throws?

A) 0.896. B) 0.80. C) 0.64. D) 0.384. Ans: A Section: 8.1 The Binomial Distributions

24. A college basketball player makes 5/6 of his free throws. Assuming free throws are

independent, the probability that he makes exactly three of his next four free throws is A)

416

3 56

1

. B) 16

3 56

1

. C) 4

16

1 56

3

. D) 16

1 56

3

. Ans: C Section: 8.1 The Binomial Distributions

25. Suppose we roll a fair die 10 times. The probability that an even number occurs exactly

the same number of times as an odd number in the 10 rolls is A) 0.1667. B) 0.2461. C) 0.3125. D) 0.5000. Ans: B Section: 8.1 The Binomial Distributions

26. A fair die is rolled 12 times. The number of times an even number occurs on the 12 rolls

has A) a binomial distribution with a mean of 2. B) a binomial distribution with a standard deviation of 3. C) a binomial distribution with a mean of 0.5. D) none of the above.


27. A college basketball player makes 80% of his free throws. Over the course of the season

he will attempt 100 free throws. Assuming free throw attempts are independent, the probability that the number of free throws he makes exceeds 80 is approximately

A) 0.2000. B) 0.2266. C) 0.5000. D) 0.7734. Ans: C Section: 8.1 The Binomial Distributions

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28. A college basketball player makes 80% of his free throws. Over the course of the season he will attempt 100 free throws. Assuming free throw attempts are independent, what is the probability that he makes at least 90 of these attempts?

A) 0.90. B) 0.72. C) 0.2643. D) 0.0062. Ans: D Section: 8.1 The Binomial Distributions

29. Which of the following random variables is geometric?

A) the number of phone calls received in a one-hour period. B) the number of cards I need to deal from a deck of 52 cards that has been thoroughly

shuffled so that at least one of the cards is a heart. C) the number of digits I will read beginning at a randomly selected starting point in a

table of random digits until I find a 7. D) all of the above.

Ans: C Section: 8.2 The Geometric Distributions


the name of each student on a 3-by-5 card. The cards are shuffled thoroughly and I choose one at random, observe the name of the student, and replace it in the set. The cards are thoroughly reshuffled and I again choose a card at random, observe the name, and replace it in the set. Let X be the number of cards observed until I get one with a name corresponding to a male student. The random variable X has which of the following probability distributions? A) the normal distribution with mean 0.5. B) the binomial distribution with p = 0.5. C) the uniform distribution. D) the geometric distribution with probability 0.5 of success.

Ans: D Section: 8.2 The Geometric Distributions


thoroughly and I turn cards over, one at a time, beginning with the top card. Let X be the number of cards I turn over until I observe the first red card. The random variable X has which of the following probability distributions? A) the normal distribution with mean 5. B) the binomial distribution with p = 0.5. C) the geometric distribution with probability of success 0.5. D) none of the above.


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32. Suppose X has a geometric distribution with probability 0.3 of success and 0.7 of failure on each observation. The probability that X is 4 is

A) 0.0081. B) 0.0189. C) 0.1029. D) 0.2401. Ans: C Section: 8.2 The Geometric Distributions

33. A college basketball player makes 80% of her free throws. Suppose this probability is

the same for each free throw she attempts. The probability that she doesn't make a free throw until her fifth attempt this season is



the same for each free throw she attempts. The probability that she makes all of her first four free throws and then misses her fifth attempt this season is

A) 0.32768. B) 0.08192. C) 0.00128. D) 0.00032. Ans: B Section: 8.2 The Geometric Distributions

35. In a test of ESP (extrasensory perception), the experimenter looks at cards that are

hidden from the subject. Each card contains either a star, a circle, a wavy line, or a square. The subject tries to read the experimenter's mind and name the shape on the cards. What is the probability that the subject gets the first four correct before giving a wrong answer if he is just guessing?

A) 0.00293. B) 0.00391. C) 0.07910. D) 0.31641. Ans: A Section: 8.2 The Geometric Distributions


the same for each free throw she attempts. The expected number of free throws required until she makes her first free throw of the season is

A) 2. B) 1.25. C) 0.80. D) 0.31. Ans: B Section: 8.2 The Geometric Distributions


hidden from the subject. Each card contains either a star, a circle, a wavy line, or a square. The subject tries to read the experimenter's mind and name the shape on the cards. If the subject is just guessing, what is the expected number of guesses before the subject gets his first correct guess?

A) 5. B) 4. C) 3. D) 2. Ans: B Section: 8.2 The Geometric Distributions

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38. Suppose X has a geometric distribution with probability 0.3 of success and 0.7 of failure on each observation. The mean and variance of X are A) mean = 0.3, variance = 0.21. C) mean = 3.33, variance = 2.79. B) mean = 1.43, variance = 0.61. D) mean = 3.33, variance = 7.77.



the same for each free throw she attempts. The probability that it takes more than three free throws before she makes her first free throw A) is 0.512. B) is 0.032. C) is 0.008. D) cannot be determined from the information given.

Ans: C Section: 8.2 The Geometric Distributions


hidden from the subject. Each card contains either a star, a circle, a wavy line, or a square. The subject tries to read the experimenter's mind and name the shape on the cards. If the subject is just guessing, what is the probability that it takes more than four trials before getting the first correct guess?

A) 0.250. B) 0.316. C) 0.422. D) 0.500. Ans: B Section: 8.2 The Geometric Distributions


the name of each student on a 3-by-5 card. The cards are shuffled thoroughly and I choose one at random, observe the name of the student, and replace it in the set. The cards are thoroughly reshuffled and I again choose a card at random, observe the name, and replace it in the set. Let X be the number of cards observed until I get one with a name corresponding to a male student. The probability that X is greater than 1 is



thoroughly and I turn cards over, one at a time, beginning with the top card. Let X be the number of cards I turn over until I observe the first red card. The probability that X is greater than 6 is

A) 1. B) 0.5. C) 0.03125. D) 0.015625. Ans: A Section: 8.2 The Geometric Distributions

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