Section 5.2 Exercises

CLARIFYING THE CONCEPTS

Question 5.107

1. Describe in your own words what it means for two events to be mutually exclusive. (p. 263)

5.2.1

Two events are mutually exclusive if they have no outcomes in common.

Question 5.108

2. Describe the intersection of two mutually exclusive events. (p. 264)

Question 5.109

3. Describe the union of two mutually exclusive events. (p. 264)

5.2.3

It is all of the outcomes in each of the events. There are no outcomes in both.

Question 5.110

4. Is it true that the union of two events always contains at least as many outcomes as the intersection of two events? Use Figures 13 and 14 to help you visualize this problem. (p. 261)

Question 5.111

5. If we choose a student at random from your college or university, is it more likely that we choose a male or a male football player? Why? (p. 261)

5.2.5

You are more likely to select a male than a male football player. All male football players are males, but most males are not football players. Therefore, there are many more males than male football players at any college or university.

Question 5.112

6. What is your personal estimate of the probability that it will rain on any given day? How about the probability that it won’t rain? Why do these numbers have to add up to 1 (or 100%)? (p. 260)

PRACTICING THE TECHNIQUES

image CHECK IT OUT!

To do Check out Topic
Exercises 7–12 Example 14 Finding the probability of
the complement of an event
Exercises 13–18 Example 15 Union and intersection
Exercises 19–44 Example 16 Addition Rule
Exercises 45–48 Example 17 Addition Rule for
mutually exclusive events

For Exercises 7–12, consider the experiment of rolling a fair die once. Find the indicated probabilities:

Question 5.113

7. Observing a number that is not 6

5.2.7

Question 5.114

8. Observing some other number than 1

Question 5.115

9. The complement of the event , where

5.2.9

Question 5.116

10. , where

Question 5.117

11. , where

5.2.11

Question 5.118

12. Not rolling an even number

For Exercises 13–18, consider the experiment of drawing a single card at random from a deck of cards. Define the following events. Find the indicated unions and intersections:

Question 5.119

13.

5.2.13

Question 5.120

14.

Question 5.121

15.

5.2.15

Question 5.122

16.

Question 5.123

17.

5.2.17

Question 5.124

18.

For Exercises 19–24, consider the experiment of drawing a single card at random from a deck of cards. Define the following events. Find the indicated probabilities:

266

Question 5.125

19.

5.2.19

Question 5.126

20.

Question 5.127

21.

5.2.21

Question 5.128

22.

Question 5.129

23.

5.2.23

Question 5.130

24.

For Exercises 25–36, refer to Table 9, a contingency table containing the counts of those who did and did not survive the sinking of the RMS Titanic, along with whether they were adults or children. Find the following probabilities:

Table 5.37: TABLE 9 Contingency table of age and survival status
Child Adult Total
Did not survive 52 1438 1490
Survived 57 654 711
Total 109 2092 2201

Question 5.131

25. (Child or Adult)

5.2.25

1

Question 5.132

26. (Child and Adult)

Question 5.133

27. (Child and Survived)

5.2.27

Question 5.134

28. (Child or Survived)

Question 5.135

29. (Child and Did not survive)

5.2.29

Question 5.136

30. (Child or Did not survive)

Question 5.137

31. (Survived and Did not survive)

5.2.31

0

Question 5.138

32. (Adult and Survived)

Question 5.139

33. (Adult or Survived)

5.2.33

Question 5.140

34. (Adult and Did not survive)

Question 5.141

35. (Adult or Did not survive)

5.2.35

Question 5.142

36. Were your chances of surviving better as an adult or as a child?

Forbes magazine's Global 2000 is a list of the 2000 largest companies on the globe. Table 10 lists the top 10 largest companies, showing the country and whether the company's assets surpass $1 trillion. Table 11 contains the contingency table of this data.

Table 5.38: TABLE 10 Top 10 companies in the Forbes Global 2000
Company Country Assets
Industrial and Commercial
Bank of China		 China Over $1 trillion
China Construction Bank China Over $1 trillion
Agricultural Bank of China China Over $1 trillion
JPMorgan Chase USA Over $1 trillion
Berkshire Hathaway USA Under $1 trillion
Exxon Mobil USA Under $1 trillion
General Electric USA Under $1 trillion
Wells Fargo USA Over $1 trillion
Bank of China China Over $1 trillion
PetroChina China Under $1 trillion
Table 5.38: Source: www.forbes.com/global2000.
Table 5.39: TABLE 11 Contingency table of country and assets
Assets
Over
$1 trillion		 Under
$1 trillion		 Total
Country China 4 1 5
USA 2 3 5
Total 6 4 10

For Exercises 37–44, the experiment is to select a company at random. Use the completed Table 10 to find the probability of the following:

Question 5.143

37. Choosing a company from China

5.2.37

Question 5.144

38. Choosing a company from the USA

Question 5.145

39. Choosing a company with over $1 trillion in assets

5.2.39

Question 5.146

40. Choosing a company with under $1 trillion in assets

Question 5.147

41. Selecting a company that is from China and has over $1 trillion in assets

5.2.41

Question 5.148

42. Selecting a company that is from the USA and has over $1 trillion in assets

Question 5.149

43. Selecting a company that is from the USA or has over $1 trillion in assets

5.2.43

Question 5.150

44. Selecting a company that is from the USA or has under $1 trillion in assets

The following table contains a list of actors who have received many Oscar nominations but have never won an Oscar.

Actor Nominations Actor Nominations
Peter O'Toole 8 Tom Cruise 3
Richard
Burton		 7 Will Smith 2
Glenn Close 6 John
Travolta		 2
Leonardo
DiCaprio		 5 Edward
Norton		 2
Sigourney
Weaver		 3 Judy
Garland		 2
Johnny Depp 3 James
Dean		 2

Suppose we select one actor at random. For Exercises 45–48, find the probability that the randomly selected actor received the following numbers of nominations without winning an Oscar:

Question 5.151

nooscars

45. 2 or 3 nominations

5.2.45

Question 5.152

nooscars

46. 2 or 4 nominations

Question 5.153

nooscars

47. 3 or 4 nominations

5.2.47

Question 5.154

nooscars

48. 7 or 8 nominations

APPLYING THE CONCEPTS

Technology Adopters. Use the following information for Exercises 49–51. The Gallup organization published a study8 in which it categorized American technology adopters as either Super Tech Adopters (31%), Smart Phone Reliants (19%), Mature Technophiles (22%), or Tech-Averse Olders (28%). Suppose we select one technology adopter at random.

267

Question 5.155

49. Find the following probabilities:

  1. Selecting a Super Tech Adopter or a Smart Phone Reliant
  2. Selecting a Super Tech Adopter or a Mature Technophile

5.2.49

(a) 0.5 (b) 0.53

Question 5.156

50. Find the following probabilities:

  1. Selecting a technology adopter who is not a Super Tech Adopter
  2. Selecting a technology adopter who is neither a Super Tech Adopter nor a Tech-Averse Older

Question 5.157

51. Find the following probabilities:

  1. Selecting a technology adopter who is a Super Tech Adopter or a Smart Phone Reliant or a Mature Technophile.
  2. Selecting a technology adopter who is neither a Super Tech Adopter nor a Smart Phone Reliant nor a Mature Technophile.

5.2.51

(a) 0.72 (b) 0.28

Question 5.158

52. Facebook Females and Males. The Facebook Social Ads Platform reported the following number of users in the United States in 2014, by gender: females: 96 million, males: 82 million, unknown: 2 million. If we choose a Facebook user at random, find the following probabilities:

  1. We choose a female or unknown.
  2. We choose a female or someone who is not a female.
  3. We select a female or a male.

Question 5.159

53. Game of Craps. You win the casino game of craps if you roll a 7 or 11. Find the probability of rolling a sum of 7 or 11 when two fair dice are rolled.

5.2.53

Card Game. Use the following experiment for Exercises 54–57: you choose one card at random from a deck of 52 cards, and you will win $3 if the card is either an ace or a heart.

Question 5.160

54. Find the probability of winning this game.

Question 5.161

55. What is a fair price to pay to play this game?

5.2.55

$0.92

Question 5.162

56. Suppose you win by drawing either a king or a red card. What is the probability of winning the game now?

Question 5.163

57. What is a fair price to pay to play the game in Exercise 56?

5.2.57

$1.62

Question 5.164

58. Social Networking Apps. The Nielsen Apps Playbook was a survey taken in 2010 of 3691 males and females on the use of social networking apps9 (mobile software applications). The results are shown in the following table:

Has downloaded a social networking
app in the last 30 days
Yes No
Male 884 841
Female 1220 746

Find the probability that a randomly chosen person has the following characteristics:

  1. Is female
  2. Has downloaded a social networking app in the last 30 days
  3. Is a female who has downloaded a social networking app in the last 30 days
  4. Is a female or has downloaded a social networking app in the last 30 days
  5. Is not a female
  6. Has not downloaded a social networking app in the last 30 days

Question 5.165

59. A study of online dating behavior found that users of a particular online dating service self-reported their physical appearance according to the counts given in Table 12.10

Table 5.42: TABLE 12 Gender and self-reported physical appearance
Gender Physical appearance
Very
attractive		 Attractive Average Prefer
not to
answer		 Total
Female 3,113 16,181 6,093 3,478 28,865
Male 1,415 12,454 7,274 2,809 23,952
Total 4,528 28,635 13,367 6,287 52,817

Using this information, find the probability that a randomly chosen online dater has the following characteristics:

  1. Is female
  2. Self-reported as attractive
  3. Is a female who self-reported as attractive
  4. Is a female or self-reported as attractive
  5. Self-reported as attractive or self-reported as average

5.2.59

(a) 0.5465 (b) 0.5422 (c) 0.3064 (d) 0.7823 (e) 0.7952

Recommended Gasoline for Different Car Sizes. Consider Table 13, showing the size and recommended gasoline for ten 2014 automobiles.

Table 5.43: TABLE 13 Size and recommended gasoline for ten 2014 automobiles
Car Car size Recommended
gasoline
BMW 328i Compact Premium
Chevrolet Camaro Compact Regular
Honda Accord Compact Regular
Cadillac CTS Midsize Premium
Nissan Sentra Midsize Regular
Subaru Legacy AWD Midsize Premium
Toyota Camry Midsize Regular
Ford Taurus Large Regular
Hyundai Genesis Large Premium
Rolls-Royce Large Premium
Table 5.43: Source: www.fueleconomy.gov.

268

Question 5.166

60. Use Table 13 to complete the contingency table shown in Table 14.

Table 5.44: TABLE 14 Contingency table of car size and recommended gasoline
Recommended gasoline
Regular Premium Total
Car Size Compact
Midsize
Large
Total

For Exercises 61–64, the experiment is to select a car at random.

Question 5.167

61. Use the completed Table 14 to find the probability of the following:

  1. Choosing a compact car
  2. Choosing a car that is not a compact car
  3. Selecting a midsize car
  4. Selecting a car that is not a midsize car

5.2.61

(a) (b) (c) (d)

Question 5.168

62. Use the completed Table 14 to find the probability of thefollowing:

  1. Choosing a car that uses regular gasoline
  2. Choosing a car that does not use regular gasoline
  3. Selecting a compact car that uses premium gasoline
  4. Selecting a compact car that uses regular gasoline

Question 5.169

63. Use the completed Table 14 to find the following probabilities:

  1. Choosing a midsize car that uses premium gasoline
  2. Choosing a midsize car that uses regular gasoline
  3. Selecting a large car that uses premium gasoline
  4. Selecting a large car that uses regular gasoline

5.2.63

(a) (b) (c) (d)

Question 5.170

64. Use the completed Table 14 to find the following probabilities:

  1. Choosing a compact car or a car that uses premium gasoline
  2. Choosing a compact car or a car that uses regular gasoline
  3. Selecting a car that uses regular gasoline and is not a compact car
  4. Selecting a car that uses regular gasoline or is not a compact car
  5. Selecting a car that uses premium gasoline or a large car

Top 10 Movies of All Time. Table 15 contains the top 10 domestic movies of all time, according to Box Office Mojo (http://boxofficemojo.com), along with the studio, whether the movie grossed over or under $500 million, and the century in which the movie was produced.

Table 5.45: TABLE 15 Top 10 domestic movies of all time
Rank Movie Studio Gross
over or
under
$500
million		 Century
produced
1 Avatar Fox Over 21st
2 Titanic Other Over 20th
3 Marvel's The
Avengers		 Other Over 21st
4 The Dark
Knight		 Warner Over 21st
5 Star Wars:
Episode I - The
Phantom Menace		 Fox Under 20th
6 Star Wars Fox Under 20th
7 The Dark
Knight Rises		 Warner Under 21st
8 Shrek 2 Other Under 21st
9 E.T.: The
Extra-Terrestrial		 Other Under 20th
10 The Hunger
Games:
Catching Fire		 Other Under 21st

For Exercises 65–69, find the following probabilities regarding the studio and what the movie grossed:

Question 5.171

65. Studio was Fox and the movie grossed over $500 million.

5.2.65

Question 5.172

66. Studio was not Fox and the movie grossed over $500 million.

Question 5.173

67. Studio was Fox or the movie grossed over $500 million.

5.2.67

Question 5.174

68. Studio was not Fox or the movie grossed over $500 million.

Question 5.175

69. Studio was Warner and the movie grossed over $500 million.

5.2.69

For Exercises 70–73, find the following probabilities about the studio and the century produced:

Question 5.176

70. Studio was Warner and the movie was produced in the 20th century.

Question 5.177

71. Studio was Warner and the movie was produced in the 21st century.

5.2.71

Question 5.178

72. Studio was not Warner and the movie was produced in the 20th century.

Question 5.179

73. Studio was not Warner and the movie was produced in the 21st century.

5.2.73

Wheel of Fortune! Use Table 16 for Exercises 74–81. Imagine yourself on the television game show Wheel of Fortune, where contestants guess the letters contained in a hidden phrase. You want to ask for the letters that have the greatest chance of occurring, so you want to know the various probabilities of the letters in the English alphabet.

The experiment is to choose one letter at random from a sample of 1000 letters. The sample space is the 26 letters of the alphabet. The total sample size is 1000, so you can find the relative frequencies of the letters in English simply by dividing each frequency by the total sample size (remember this from Chapter 2?).

269

Table 5.46: TABLE 16 Frequency distribution of English letters
A B C D E F G
73 9 30 44 130 28 16
H I J K L M N
35 74 2 3 35 25 78
O P Q R S T U
74 27 3 77 63 93 27
V W X Y Z
13 16 5 19 1

Question 5.180

74. Construct a probability model of the frequency distribution in Table 16.

Question 5.181

75. Find the probability of selecting an E.

5.2.75

0.13

Question 5.182

76. Find the probability of selecting a vowel, that is, either A, E, I, O, or U.

Question 5.183

77. Note that selecting a consonant may be considered the complement of selecting a vowel. Find the probability of selecting a consonant.

5.2.77

0.622

Question 5.184

78. Choosing one letter at random, which is it more likely to be, a consonant or a vowel?

Question 5.185

79. Of the eight letters with the highest frequencies, how many are vowels?

5.2.79

4

Question 5.186

80. Compare your answers to Exercises 76 and 77, and explain why Wheel of Fortune contestants are not allowed to guess vowels (they have to buy them).

Question 5.187

81. If you were a contestant on Wheel of Fortune and had no money to buy a vowel, what would be your first five letter choices?

5.2.81

T, N, R, S, D

Gender and GPA. The National Center for Education Statistics reported the contingency table results in Table 17 for the gender and the grade point average of the 21,504 undergraduate students it surveyed. The experiment is to select one random selected student. Use Table 17 to calculate the probabilities indicated in Exercises 82–93.

Table 5.47: TABLE 17 Contingency table of gender and GPA
GPA < 2.5 GPA 2.5
to < 3		 GPA 3
to < 3.5		 GPA
3.5 or
higher		 Total
Females 2,698 2,439 3,314 3,868 12,319
Males 2,535 1,975 2,324 2,351 9,185
Total 5,233 4,414 5,638 6,219 21,504

Question 5.188

82. Student is female.

Question 5.189

83. Student is male.

5.2.83

0.4271

Question 5.190

84. Student has .

Question 5.191

85. Student is female and has a .

5.2.85

0.1255

Question 5.192

86. Student is male and has a .

Question 5.193

87. Student is female or has a .

5.2.87

0.6908

Question 5.194

88. Student is male or has .

Question 5.195

89. Student does not have a .

5.2.89

0.7566

Question 5.196

90. Student is female and does not have a .

Question 5.197

91. Student is male and does not have a .

5.2.91

0.3092

Question 5.198

92. Student is female or does not have a .

Question 5.199

93. Student is male or does not have a .

5.2.93

0.8745

BRINGING IT ALL TOGETHER

Best-Selling Video Games. Table 18 contains the top 20 best-selling video games in the United States for the week of May 17, 2014, along with the game platform, publishing studio, and the type of game. Use this information for Exercises 94–102.

Table 5.48: TABLE 18 Top 20 best-selling video games in the United States for the week of May 17, 2014
Game/Platform Studio Type
Minecraft for PS3 Sony Adventure
Minecraft for Xbox 360 MS Adventure
Kirby: Triple Deluxe for 3DS Nintendo Platform
MLB 14 The Show for PS4 Sony Sports
Titanfall for Xbox One Electronic
Arts		 Shooter
Call of Duty: Ghosts for
Xbox 360		 Activision Shooter
Bound by Flame for PS4 Focus Action
Pokemon X/Y for 3DS Nintendo Role-Playing
Titanfall for Xbox 360 Electronic
Arts		 Shooter
Grand Theft Auto V for
Xbox 360		 Take-Two Action
Grand Theft Auto V for PS3 Take-Two Action
Call of Duty: Ghosts for PS4 Activision Shooter
Super Luigi U for WiiU Nintendo Platform
Super Mario Brothers U for
WiiU		 Nintendo Action
Call of Duty: Ghosts for PS3 Activision Shooter
Borderlands 2 for PSV Take-Two Shooter
Battlefield 4 for Xbox 360 Electronic
Arts		 Shooter
Forza Motorsport 5 for
Xbox One		 MS Racing
Call of Duty: Ghosts for
Xbox One		 Activision Shooter
inFamous: Second Son for PS4 Sony Action

Question 5.200

94. Suppose our experiment is to select one video game at random from Table 18, and observe its studio. List the video games belonging to the following events:

  1. S: Sony
  2. N: Nintendo.

Question 5.201

95. For each studio in Exercise 94, find the probability of a video game being made by that studio.

5.2.95

Studio Probability
Sony
MS
Nintendo
Electronic Arts
Activision
Focus
Take-Two

270

Question 5.202

96. Describe in words the complement of the followingevents (save space by not listing them all):

  1. S: Sony
  2. N: Nintendo.

Question 5.203

97. For each complement in Exercise 96, find its probability.

5.2.97

(a) (b)

Question 5.204

98. What is the sum of the following two probabilities: (i) Video game made by Sony, (ii) Video game made by some other studio.

Question 5.205

99. Next, suppose our experiment is to select a video game at random. Define the following events: S: Studio is Sony, N: Studio is Nintendo, A: Type = Action, P: Type = Platform. List the video games in the following events:

  1. (S and A)
  2. (S or A)
  3. (N and P)
  4. (N or P)
  5. (N and A)
  6. (N or A)

5.2.99

(a) inFamous: Second Son for PS4 (b) Minecraft for PS3, MLB 14 The Show for PS4, inFamous: Second Son for PS4, Super Mario Brothers U for WiiU, Grand Theft Auto V for PS3, Grand Theft Auto V for Xbox 360, Bound by Flame for PS4 (c) Kirby: Triple Deluxe for 3DS, Super Luigi U for WiiU (d) Kirby: Triple Deluxe for 3DS, Pokeman X/Y for 3DS, Super Luigi U for WiiU, Super Mario Brothers U for WiiU (e) Super Mario Brothers U for WiiU (f) Kirby: Triple Deluxe for 3DS, Pokeman X/Y for 3DS, Super Luigi U for WiiU, Super Mario Brothers U for WiiU, inFamous: Second Son for PS4, Grand Theft Auto V for PS3, Grand Theft Auto V for Xbox 360, Bound by Flame for PS4

Question 5.206

100. Find the following probabilities:

  1. (S and A)
  2. (S or A)
  3. (N and P)
  4. (N or P)
  5. (N and A)
  6. (N or A)

Question 5.207

101. List the video games in the following events:

  1. (S and N)
  2. (S or N)
  3. (A and P)
  4. (A or P)

5.2.101

(a) None (b) Minecraft for PS3, MLB 14 The Show for PS4, inFamous: Second Son for PS4, Kirby: Triple Deluxe for 3DS, Pokeman X/Y for 3DS, Super Luigi U for WiiU, Super Mario Brothers U for WiiU (c) None (d) inFamous: Second Son for PS4, Grand Theft Auto V for PS3, Grand Theft Auto V for Xbox 360, Bound by Flame for PS4, Kirby: Triple Deluxe for 3DS, Super Luigi U for WiiU, Super Mario Brothers U for WiiU

Question 5.208

102. Find the following probabilities:

  1. (S and N)
  2. (S or N)
  3. (A and P)
  4. (A or P)

WORKING WITH LARGE DATA SETS

image Chapter 5 Case Study: The Gardasil Vaccine. Open the data set Gardasil. We will explore some probabilities about the patient completion of the vaccination treatment and whether the patient received medical assistance, using the tools and techniques we have learned in this section. Use technology to complete Exercises 103–107.

Question 5.209

gardasil

103. Construct a relative frequency distribution of the variables medical assistance and completed. Use these to find the probability that a randomly selected patient:

  1. had medical assistance.
  2. completed the vaccine treatment.

5.2.103

Had Medical Assistance Frequency Relative Frequency
No 1138 0.8054
Yes 275 0.1946
Total 1413 1.0000
Completed Medical Treatment Frequency Relative Frequency
No 944 0.6681
Yes 469 0.3319
Total 1413 1.0000

(a) 0.1946

(b) 0.3319

Question 5.210

gardasil

104. Construct a contingency table (crosstabulation) of medical assistance with completed. Include the row totals, column totals, and grand total.

Question 5.211

gardasil

105. Use the contingency table from Exercise 104 to find the probability that a randomly chosen patient:

  1. had medical assistance and completed the treatment.
  2. had medical assistance and did not complete the treatment.
  3. did not have medical assistance and did complete the treatment.
  4. did not have medical assistance and did not complete the treatment.

5.2.105

(a) 0.0389 (b) 0.1557 (c) 0.2930 (d) 0.5124

Question 5.212

gardasil

106. Think about the four probabilities you found in Exercise 105. Do you think that patients who had medical assistance were more likely to complete the treatment? Provide support for your statement.

Question 5.213

gardasil

107. Find the probability that a randomly chosen patient:

  1. had medical assistance or completed the treatment.
  2. had medical assistance or did not have medical assistance.

5.2.107

(a) 0.4876 (b) 1