1. State the four requirements for a binomial experiment. (p. 327)

2. What is meant by a “success” in a binomial experiment? Is a success always a good thing? (p. 328)

3. In a binomial experiment, explain why it is not possible for to exceed . (p. 327)

4. Restate the binomial probability distribution formula using the following terms: , the probability of success, the number of successes, the probability of failure, and the number of failures. (p. 330)

5. Ask 10 of your friends to come to your party (remember the independence assumption on page 327).

6. Toss a fair die three times, and note the total number of spots.

7. Answer a random sample of eight multiple-choice questions either correctly or incorrectly by random guessing. There are four choices, (a)-(d), for each question.

8. Toss a fair die three times, and note the number of 6s.

9. Select a student at random in the class until you come across a left-handed student.

10. Four cards are selected at random with replacement from a deck of cards, and the number of queens is observed.

11. Four cards are selected at random without replacement from a deck of cards, and the number of queens is observed.

12. Four cards are selected at random with replacement from a deck of cards, and the total number of blackjack-style points (number cards = number of points; face cards = 10 points; aces = either 1 or 11) is calculated.

13. Bob has paid to play two games at a carnival. The probability that he wins a particular game is 0.25.

14. Bob is playing a game at a carnival where he gets to play until he loses. The probability that he wins a particular game is 0.25.

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26. (Hint: Use the result from Exercise 25.)

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29. Observe no business majors

30. Observe one business major

31. Observe two business majors

32. Observe at most two business majors

33. Observe at least one business major

34. Observe between zero and two business majors, inclusive

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41. None are living alone.

42. At least one is living alone.

43. At most two are living alone.

44. Six are living alone.

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49. Business majors accounted for 25% of the proportion of all Master's degrees granted in 2012. Select three Master's degrees at random. Let X = the number of business majors.

50. The CDC found that 7.5% of children ages 6-17 used prescribed medication during the past six months for emotional or behavioral difficulties. Select four children ages 6-17 at random, and let X = the number of children using prescription medication for emotional or behavioral difficulties.

51. Ten percent of Americans ages 25–29 live alone. Take a random sample of five Americans ages 25-29, and observe X = the number living alone.

52. Forty percent of 18- to 29-year-olds have a Twitter account. Take a random sample of six 18- to 29-year-olds, and find X = the number who have a Twitter account.

53. The binomial experiment in Exercise 49

54. The binomial experiment in Exercise 50

55. The binomial experiment in Exercise 51

56. The binomial experiment in Exercise 52

57. Random Guessing on a Quiz. Suppose that you are taking a quiz of five multiple-choice questions (the instructor chose the questions randomly), with each question having four possible responses. You did not study at all for the quiz and will randomly guess the correct response for each question. The random variable is the number of correct responses.

58. Women in Management. According to the U.S. Government Accountability Office, women hold 40% of the management positions in the United States.³ Suppose we take a random sample of 20 people in management positions.

59. Abandoning Landlines. The Centers for Disease Control reported in 2014 that 41% of U.S. households use only cell phones (no landline) (Source: http://www.pewresearch.org/fact-tank/2014/07/08/two-of-every-five-u-s-households-have-only-wireless-phones/). Suppose we take a random sample of 12 telephone users.

60. Online Dating. The Pew Research Center reported in 2014 that 22% of 25- to 34-year-olds have used online dating. Suppose we select fifteen 25- to 34-year-olds at random.

61. Random Guessing on a Quiz. Refer to Exercise 57.

62. Women in Management. Refer to Exercise 58.

63. Abandoning Landlines. Refer to Exercise 59.

64. Online Dating. Refer to Exercise 60.

65. Women and Depression. According to the National Institute of Mental Health, nearly twice the proportion of women (12%) as men (6.6%) are affected by a depressive disorder each year. Suppose that random samples of five women and five men are taken. Let represent the number of women affected by a depressive disorder.

66. Men and Depression. Refer to Exercise 65. Let represent the number of men affected by a depressive disorder in a random sample of size 5.

67. Mean, Median, Mode. For a binomial distribution, if the mean is a whole number, then . Use this equation to answer the following questions:

68. Geometric Probability Distribution. Refer to Example 14(a), where a fisherman is going fishing and will continue to fish until he catches a rainbow trout. This is an example of the geometric probability distribution, which has the same requirements as the binomial distribution, except that there is not a fixed number of trials . Instead, the geometric random variable represents the number of trials until a success is observed. The geometric probability distribution formula is

where represents the probability of success. The possible values of are . The U.S. Census Bureau reported in 2010 that 30% of U.S. households have no access at all to the Internet. A random sample is taken of U.S. households. Let the random variable represent the number of trials until a household is found that has access to the Internet.

69. Hypergeometric Probability Distribution. If samples are drawn from a relatively small finite population, and the sample size is larger than 1% of the population, so that the 1% Guideline (page 282) does not apply, we should not use the binomial distribution because the samples are not independent. Instead, if we are sampling without replacement, and there are two mutually exclusive categories, then you should use the hypergeometric probability distribution. Suppose that objects belong to the first category (“successes”), and objects belong to the second category (“failures”). Then the probability of getting successes and failures is given by the hypergeometric probability distribution formula:

where , is the population size, and is the sample size. You are dealt 5 cards at random from a deck of 52 cards.

70. Multinomial Distribution. The multinomial probability distribution is similar to the binomial distribution, except that the binomial involves only two categories, whereas the multinomial involves more than two categories. Suppose we have three mutually exclusive outcomes, , and , where , , and . If we have a sample of independent trials, then the probability that we get outcomes of category , outcomes of category , and outcomes of category is given by the following formula:

Suppose that 30% of students on a particular college campus are Democrats, 30% are Republicans, and 40% are Independents. Suppose we take a random sample of 10 students.

71. Confirm that this situation represents a binomial experiment.

72. Use the binomial distribution formula to find the following probabilities:

73. Use the binomial tables or technology to find the following probabilities. Then explain why the probabilities in (a) and (b) are equal, as are the probabilities in (c) and (d).

74. Find the following parameters of the binomial distribution for this experiment. Interpret the mean and standard deviation.

75. If possible, find the probability that equals . If not possible, explain why it is not possible to do so.

76. Construct the probability distribution graph of .

77. Identify the mode of .

78. Find the probability that equals the mode of .

79. Following Example 19(c), identify all values of this binomial distribution that are unusual or somewhat unusual.

80. Examine the variable class. Make this into a binomial random variable as follows: Consider all vehicles that are compact cars to be a success, and all other vehicles to be a failure. If we use all 1141 vehicles, what is the probability of success?

81. If we take a random sample of 100 vehicles, what is the expected number of compact cars?

82. If we take a random sample of 100 vehicles, what is the standard deviation of the number of compact cars?

83. Use technology to obtain a random sample of 100 vehicles. How many compact cars are in the sample?

84. Use the population mean and standard deviation from Exercises 81 and 82 to determine whether the observed number of compact cars in your sample in Exercise 83 is unusual, using the -score method.