Section 5.3 Exercises

CLARIFYING THE CONCEPTS

286

Question 5.214

1. Suppose you are the coach of a football team, and your star quarterback is injured. (p. 274)

  1. Does the injury affect the chances that your team will win the big game this weekend?
  2. How would you describe this situation in the terminology presented in this section?

5.3.1

(a) Yes. (b) The probability of winning the football game depends on whether or not the star quarterback can play in the game.

Question 5.215

2. Write a sentence or two about a situation in your life similar to Exercise 1, where the probability of some event was affected by whether or not some other event occurred. (p. 274)

Question 5.216

3. Explain clearly the difference between and. (p. 272)

5.3.3

For , we assume that the event has occurred, and now need to find the probability of event , given event . On the other hand, for , we do not assume that event has occurred, and instead need to determine the probability that both events occurred.

Question 5.217

4. Give an example from your own experience of two events that are independent. Describe how they are independent. (p. 274)

Question 5.218

5. Picture yourself explaining the Gambler's Fallacy to your friends. How would you explain the Gambler's Fallacy in your own words? (p. 275)

5.3.5

Answers will vary.

Question 5.219

6. Explain why two events, and , cannot have the following characteristics: . (p. 272)

Question 5.220

7. Explain why each of the following events is either dependent or independent: (p. 277)

  1. Drawing a ball from a box, replacing it, and then drawing a second ball
  2. Drawing a ball from a box, not replacing it, and then drawing a second ball

5.3.7

(a) Independent; sampling with replacement (b) Dependent; sampling without replacement

Question 5.221

8. Explain why the following events are either dependent or independent, and provide support for your assertion: (p. 277)

  1. Tossing a coin and drawing a card from a deck of playing cards
  2. Drawing a card from a deck, not replacing it, and drawing another card

PRACTICING THE TECHNIQUES

image CHECK IT OUT!

To do Check out Topic
Exercises 9–32 Example 19 Conditional probability
Exercises 33–52 Example 20 Determining whether
two events are
independent
Exercises 53–58 Example 22 Multiplication Rule
Exercises 59–66 Example 23 Multiplication Rule for
two independent events
Exercises 67–70 Example 24 Sampling with
replacement
Exercises 71–74 Example 25 Sampling without
replacement
Exercises 79–84 Example 26 Determining
independence using the
alternative method
Exercises 85–92 Example 27 Conditional probability
for mutually exclusive
events
Exercises 93–100 Example 28 Multiple rule for
independent events
Exercises 100–104 Example 29 Solving an “at least”
problem
Exercises 75–78 Example 30 Applying the 1%
Guideline
Exercises 105–108 Example 31 Bayes' Rule

Table 26 presents a sample of 10 threatened and endangered mammals, and their continents. The contingency table for this data is given in Table 27.

Table 5.60: TABLE 26 Endangered and threatened mammals
Species Continent Endangered or
threatened?
Desert Bandicoot Australia Endangered
Grizzly Bear North America Threatened
Chimpanzee Africa Endangered
African elephant Africa Threatened
Koala Australia Threatened
Mountain gazelle Africa Endangered
Canada Lynx North America Threatened
Ocelot North America Endangered
Bighorn sheep North America Endangered
White rhino Africa Endangered
Table 5.61: TABLE 27 Contingency table of threatened and endangered species
Africa Australia North
America		 Total
Threatened 1 1 2 4
Endangered 3 1 2 6
Total 4 2 4 10

A mammal is to be chosen at random. Define the following events:

  • A: Continent is Africa
  • B: Continent is Australia
  • C: Continent is North America
  • E: Mammal is endangered
  • T: Mammal is threatened

For Exercises 9–22, use Table 27 to find the indicated probabilities.

287

Question 5.222

9. The mammal is from Africa, given that it is endangered, (A given E)

5.3.9

Question 5.223

10. (B given E)

Question 5.224

11. (C given E)

5.3.11

Question 5.225

12. (A given T)

Question 5.226

13. (B given T)

5.3.13

Question 5.227

14. (C given T)

Question 5.228

15. The mammal is endangered, given that it is from Africa, (E given A)

5.3.15

Question 5.229

16. (E given B)

Question 5.230

17. (E given C)

5.3.17

Question 5.231

18. (T given A)

Question 5.232

19. (T given B)

5.3.19

Question 5.233

20. (T given C)

Question 5.234

21. Compare (E given A) and (T given A). Among the mammals from Africa, is the higher proportion endangered or threatened?

5.3.21

Endangered

Question 5.235

22. Compare (A given E), (B given E), and (C given E). Among the endangered animals, which continent has the highest proportion of mammals?

Use the contingency table in Table 28 to find the indicated probabilities For Exercises 23–26. A student is chosen at random. Define the following events:

  • U: Undergrad, G: Graduate student, D: Prefers desktops for email, S: Prefers smartphones for email.
Table 5.62: TABLE 28 Contingency table of D and S versus U and G
Desktops Smartphones Total
Undergrads 2 4 6
Graduate students 3 6 9
Total 5 10 15

Question 5.236

23. Student prefers desktops for email, given student is an undergrad, (D given U)

5.3.23

Question 5.237

24. (D given G)

Question 5.238

25. (S given U)

5.3.25

Question 5.239

26. (S given G)

Use the contingency table in Table 29 to find the indicated probabilities for Exercises 27–32. A person is chosen at random. Define the following events:

  • U: Undergrad, G: Graduate student, P: Parent, D: Prefers desktops for email, S: Prefers smartphones for email.
Table 5.63: TABLE 29 Contingency table of D and S versus U, G, and P
Desktops Smartphones Total
Undergrads 2 4 6
Graduate students 3 6 9
Parents 5 0 5
Total 10 10 20

Question 5.240

27. Student prefers desktops for email, given student is an undergrad, (D given U)

5.3.27

Question 5.241

28. (D given G)

Question 5.242

29. (D given P)

5.3.29

1

Question 5.243

30. (S given U)

Question 5.244

31. (S given G)

5.3.31

Question 5.245

32. (S given P)

For Exercises 33–38, refer to your work in Exercises 9–22. Use the strategy for determining whether two events are independent (page 274) to determine whether the following pairs of events are independent:

Question 5.246

33. Species is from Africa, and the species is endangered, A and E

5.3.33

Not independent

Question 5.247

34. B and E

Question 5.248

35. C and E

5.3.35

Not independent

Question 5.249

36. A and T

Question 5.250

37. B and T

5.3.37

Not independent

Question 5.251

38. C and T

For Exercises 39–42, refer to your work in Exercises 23–26. Use the strategy for determining whether two events are independent (page 274) to determine whether the following pairs of events are independent:

Question 5.252

39. D and U

5.3.39

Independent

Question 5.253

40. D and G

Question 5.254

41. S and U

5.3.41

Independent

Question 5.255

42. S and G

For Exercises 43–48, refer to your work in Exercises 27–32. Use the strategy for determining whether two events are independent (page 274) to determine whether the following pairs of events are independent:

Question 5.256

43. D and U

5.3.43

Not independent

Question 5.257

44. D and G

Question 5.258

45. D and P

5.3.45

Not independent

Question 5.259

46. S and U

Question 5.260

47. S and G

5.3.47

Not independent

Question 5.261

48. S and P

Question 5.262

49. Refer to Exercises 33–38. How would you respond if someone claimed that whether a species was threatened or endangered did not depend on the continent from which the species came?

5.3.49

It is not true.

Question 5.263

50. In Exercise 49, which probabilities would you use to support your argument?

Question 5.264

51. Refer to Exercises 39–42. Suppose someone claimed that whether a student preferred a desktop or a smartphone for email depended on whether the student was an undergrad or graduate student. How would you respond?

5.3.51

It is not true.

Question 5.265

52. Refer to Exercises 43–48. Suppose someone claimed that whether a person preferred a desktop or a smartphone for email depended on whether the person was an undergrad or a parent. How would you respond?

For Exercises 53–58, use the Multiplication Rule to find the indicated probability.

Question 5.266

53. The National Center for Education Statistics reports that 71% of college students applied for federal financial aid. Of those who applied for federal aid, 47.5% were male. Choosing a student at random, what is the probability of selecting a male who applied for federal financial aid?

5.3.53

0.33725

288

Question 5.267

54. Of all the police precincts in New York City, 32.89% are in Brooklyn. Of the precincts in Brooklyn, 56% are in South Brooklyn. Find the probability of selecting a police precinct in South Brooklyn.

Question 5.268

55. Thirty percent of students at a particular college take statistics. Ninety percent of students taking statistics at the college pass the course. What is the probability that a student will take statistics and pass the course?

5.3.55

0.27

Question 5.269

56. Fifty percent of students at a particular college are commuters. Of those, 10% bike to school. Find the probability that a student is a commuter and bikes to school.

Question 5.270

57. Twenty-five percent of the nursing students at a particular college are male. Of these, 50% are taking a biology course this semester. Calculate the probability that a nursing student is a male and is taking a biology course this semester.

5.3.57

0.125

Question 5.271

58. Thirty percent of the statistics students at a particular college have taken advantage of the college tutoring program. After doing so, 80% of them received a higher score on the next exam. Find the probability that a statistics student has taken advantage of the college tutoring program and has received a higher score on the next exam.

The National Center for Education Statistics reports the following statistics for surveys of 12,320 female college students and 9,184 male college students:

  • 36% of females work 16–25 hours per week
  • 38% of males work 16–25 hours per week

A random sample of college students is taken. Define the following events:

  • F1: 1st draw is a female who works 16–25 hours
  • F2: 2nd draw is a female who works 16–25 hours
  • M1: 1st draw is a male who works 16–25 hours
  • M2: 2nd draw is a male who works 16–25 hours

For Exercises 59–66, use the Multiplication Rule for Independent Events to calculate the indicated probabilities:

Question 5.272

59. (F1 and F2)

5.3.59

0.1296

Question 5.273

60. (F1 and M2)

Question 5.274

61. (M1 and F2)

5.3.61

0.1368

Question 5.275

62. (M1 and M2)

Question 5.276

63. (F2 given F1)

5.3.63

0.36

Question 5.277

64. (M2 given F1)

Question 5.278

65. Compare (F1 and F2) with (F2 given F1). Why is this so?

5.3.65

(F1 and F2) = (0.36)(0.36) whereas (F2 given F1) = 0.36; sampling less than 1% of population so 1% Guideline applies.

Question 5.279

66. Compare (F1 and M2) with (M2 given F1). Why is this so?

For Exercises 67–70, suppose we sample two cards at random and with replacement from a deck of cards. Define the following events: : Red card observed on the first draw, : Red card observed on the second draw, : Heart observed on the first draw, : Heart observed on the second draw. Find the following probabilities:

Question 5.280

67.

5.3.67

0.25

Question 5.281

68.

Question 5.282

69.

5.3.69

0.125

Question 5.283

70.

For Exercises 71–74, suppose we sample three cards at random and without replacement from a deck of cards. Define the same events as for Exercises 67–70, and define event as Red card observed on the third draw, and define event as Heart observed on the third draw.

Question 5.284

71.

5.3.71

0.2451

Question 5.285

72.

Question 5.286

73.

5.3.73

0.1176

Question 5.287

74.

Use the following information for Exercises 75–78. The College Board reports that 48% of the 1.5 million students who took the Natural Sciences subject exam in 2014 had taken 4 years of high school science.

Question 5.288

75. If we take a sample of the following numbers of students, verify that the 1% Guideline applies:

5.3.75

(a) A sample of size is of the population, which is less than 1%. Therefore the 1% Guideline applies. (b) A sample of size of the population, which is less than 1%. Therefore the 1% Guideline applies. (c) A sample of size is of the population, which is less than 1%. Therefore the 1% Guideline applies.

For Exercises 76–78, use the 1% Guideline to approximate the probability that the indicated number of students who took the Natural Sciences subject exam had taken four years of high school science.

Question 5.289

76.

Question 5.290

77.

5.3.77

0.1106

Question 5.291

78.

Table 30 represents the New York Times Best-Seller List for manga (a type of graphic novel), for the week of August 24, 2014. Table 31 contains a contingency table of this data, for the publisher and the weeks on the bestseller list. Use this information for Exercises 79–88.

Table 5.64: TABLE 30 New York Times Best-Seller List for manga, week of August 24, 2014
Manga title Publisher Weeks
on list
Crimson Spell VIZ Media 1
Bleach VIZ Media 2
Deadman Wonderland VIZ Media 1
Fairy Tail Kodansha 1
Loveless VIZ Media 1
Food Wars! VIZ Media 2
Attack on Titan: No Regrets Kodansha 8
Rosario + Vampire VIZ Media 2
Dragonar Academy Seven Seas 2
Attack on Titan Kodansha 62
Table 5.64: Source: NYTimes.com.

289

Table 5.65: TABLE 31 Contingency table of manga publisher and weeks on best-seller list
VIZ Media Kodansha Seven Seas Total
1 Week 3 1 0 4
2 Weeks 3 0 1 4
>2 Weeks 0 2 0 2
Total 6 3 1 10

A manga title is to be selected at random. Define the following events:

  • V: Manga title is published by VIZ Media
  • K: Manga title is published by Kodansha
  • A: Manga title has been on list for one week.
  • B: Manga title has been on list for two weeks.
  • C: Manga title has been on list for more than two weeks.

For Exercises 79–84, use the alternative method for determining independence (page 279) to determine whether the following pairs of events are independent:

Question 5.292

79. V and A

5.3.79

Not independent

Question 5.293

80. K and A

Question 5.294

81. V and B

5.3.81

Not independent

Question 5.295

82. K and B

Question 5.296

83. V and C

5.3.83

Not independent

Question 5.297

84. K and C

For Exercises 85–88, determine whether the following events are independent. Note that the events are mutually exclusive:

Question 5.298

85. V and K

5.3.85

Not independent

Question 5.299

86. A and B

Question 5.300

87. A and C

5.3.87

Not independent

Question 5.301

88. B and C

Question 5.302

89. Suppose , for events . State whether X and Y are independent.

5.3.89

Not independent

Question 5.303

90. Define the following events: or more, . Are A and B independent? Why?

Question 5.304

91. The intersection between events is empty. Then is it true or not true that ? Explain.

5.3.91

No; they are mutually exclusive

Question 5.305

92. Define event : team wins. Are and independent? Why?

For Exercises 93–96, use the Multiplication Rule for Independent Events to find the probabilities. Define : observe an even number on a toss of a fair die.

Question 5.306

93. occurs on three successive tosses.

5.3.93

Question 5.307

94. occurs on four successive tosses.

Question 5.308

95. occurs on five successive tosses.

5.3.95

Question 5.309

96. occurs on 10 successive tosses.

For Exercises 97–100, define : observe a number greater than 3 on a toss of a fair six-sided die. Find the following probabilities:

Question 5.310

97. That occurs at least once in three tosses

5.3.97

Question 5.311

98. That occurs at least once in four tosses

Question 5.312

99. That occurs at least once in five tosses

5.3.99

Question 5.313

100. That occurs at least once in 10 tosses

Google reports that 50% of incoming emails to Gmail are encrypted.13 For the random samples of size emails in Exercises 101–104, find the probability that at least one of the emails is encrypted.

Question 5.314

101.

5.3.101

0.75

Question 5.315

102.

Question 5.316

103.

5.3.103

0.9375

Question 5.317

104.

For Exercises 105–108, use Bayes' Rule to find (A | B) for the indicated probabilities.

Question 5.318

105.

5.3.105

0.375

Question 5.319

106.

Question 5.320

107.

5.3.107

0.2941

Question 5.321

108.

APPLYING THE CONCEPTS

Question 5.322

109. Mobile YouTube Watchers. YouTube reports that 40% of users access YouTube with a mobile device.

  1. Find the probability that two randomly selected YouTube watchers are using mobile devices.
  2. Find the probability that three randomly selected YouTube watchers are all using mobile devices.
  3. Find the probability that five randomly selected YouTube watchers are all using mobile devices.
  4. Find the probability that at least one of four randomly selected YouTube watchers is using a mobile device.

5.3.109

(a) 0.16 (b) 0.064 (c) 0.01024 (d) 0.8704

Question 5.323

110. China's Groundwater Problem. The Web site http://chinawaterrisk.org reports that 60% of China's underground water sources are polluted.

  1. Find the probability that two randomly selected underground water sources are polluted.
  2. Find the probability that three randomly selected underground water sources are polluted.
  3. Find the probability that at least one of three randomly selected underground water sources is polluted.

Question 5.324

111. Social Media and Video Games. A study14 reported that 60% of social media users play video games. Suppose this study was based on 10 social media users, and we sample three of these without replacement. Find the following probabilities:

  1. The first social media user plays video games.
  2. The second social media user plays video games, given that the first social media user plays video games.
  3. The third social media user plays video games, given that the first two social media users play video games.

5.3.111

(a) 0.6 (b) 5/9 (c) 1/2

Associate's Degree or Bachelor's Degree? Use the following information for Exercises 112–114: In its survey of undergraduate college students, the National Center for Educational Statistics reported the frequencies in the following contingency table for the number of students in an Associate's degree program and the number of students in a Bachelor's degree program, along with the student's gender. One student is selected at random.

290

Associate's Bachelor's Total
Female 5,276 5,491 10,767
Male 3,819 4,483 8,302
Total 9,095 9,974 19,069

Question 5.325

112. Find the following probabilities:

  1. Student is in an Associate's degree program.
  2. Student is in a Bachelor's degree program.
  3. Student is female.
  4. Student is male.

Question 5.326

113. Calculate the following conditional probabilities:

  1. Student is in an Associate's degree program, given that the student is female.
  2. Student is in an Associate's degree program, given that the student is male.
  3. Student is in a Bachelor's degree program, given that the student is female.
  4. Student is in a Bachelor's degree program, given that the student is male.

5.3.113

(a) 0.4900 (b) 0.4600 (c) 0.5100 (d) 0.5400

Question 5.327

114. Determine whether the following events are independent:

  1. Being in an Associate's Degree program, and being female
  2. Being in an Associate's degree program, and being male
  3. Being in a Bachelor's degree program, and being female
  4. Being in a Bachelor's degree program, and being male

More Millenials in College. The following contingency table contains the results of a survey by the Pew Research Center comparing the number of Millennials in 2013 who completed a Bachelor's degree versus the number of Generation X'ers in 1995 who completed a Bachelor's degree. Use this information for Exercises 115–118.

Completed
Bachelor's
degree or
more		 Completed
less than a
Bachelor's
degree		 Total
Millennials: 2013 6,686 11,662 18,348
Gen X'ers: 1995 4,458 10,897 15,355
Total 11,144 22,559 33,703

Question 5.328

115. Find the probability that a randomly chosen survey respondent has the following characteristics:

  1. Completed a Bachelor's degree or more, .
  2. Is a Millennial, .

5.3.115

(a) 0.3307 (b) 0.5444

Question 5.329

116. Find the following probabilities for a randomly selected survey respondent:

Question 5.330

117. Find the following conditional probabilities for a randomly chosen survey respondent:

5.3.117

(a) 0.3644 (b) 0.6356 (c) 0.2903 (d) 0.7097

Question 5.331

118. Based on your answers to the three previous exercises, state whether completing a Bachelor's degree is independent of being a Millennial.

Happiness in Marriage. Use the following information for Exercises 119–121: The General Social Survey tracks trends in American society through annual surveys. The married respondents were asked to characterize their feelings about being married. The results, crosstabulated with gender, are shown in the following table.

Happiness of marriage Total
Very
happy		 Pretty
happy		 Not too
happy
Gender Male 242 115 9 366
Female 257 149 17 423
Total 499 264 26 789

Question 5.332

119. Find the probabilities that a randomly chosen person has the following characteristics:

  1. Is female,
  2. Is male,
  3. Is not too happily married,

5.3.119

(a) 0.5361 (b) 0.4639 (c) 0.0330

Question 5.333

120. Find the probabilities that a randomly chosen person has the following characteristics:

  1. Is female and not too happily married,
  2. Is male and not too happily married,

Question 5.334

121. Are gender and being not too happily married independent? Why or why not?

5.3.121

Not independent.

Question 5.335

122. Acceptance Sampling. You are in charge of purchasing for a large computer retailer. Your wholesaler delivers computers to you in batches of 100. You either accept or reject an entire batch based on a random sample of two computers: if both computers you sample are defective, then you reject the entire batch. Suppose that (unknown to you, of course) there are 10 defective computers in the batch of 100 computers.

  1. Should you conduct your sampling with or without replacement? Why?
  2. What is the probability that the first computer you select is defective?
  3. What is the probability that the second computer you select is defective, given that the first was defective, if you sample without replacement?
  4. What is the probability that you will accept the batch?
  5. What is the probability that you will reject the batch?

Question 5.336

123. Acceptance Sampling. Refer to the previous exercise. Usually, you accept each batch of computers from this wholesaler.

  1. Do you think that is a wise move, considering that 10% of their product is defective?
  2. How could you make your test stricter so that there is a smaller chance of accepting a batch with 10% defectives?

5.3.123

Either reject the batch if at least one computer is defective or increase the sample size.

291

BRINGING IT ALL TOGETHER

image Vaccine Completion and Practice Type. Case study: The Gardasil Vaccine. Here is a contingency table of those who completed the vaccine treatment and the (medical provider's) practice type. Use this information for Exercises 124–127.

Practice type
Pediatric Family OB/
GYN		 Total
Completed No 353 259 332 944
Yes 162 106 201 469
Total 515 365 533 1413

Question 5.337

124. Find the probability that a randomly chosen patient has the following characteristics:

  1. Visited a family-type practice,
  2. Visited an OB/GYN-type practice,
  3. Completed the treatment,

Question 5.338

125. Find the probabilities that a randomly chosen patient has the following characteristics:

  1. Visited a family practice and completed the treatment
  2. Visited an OB/GYN practice and completed the treatment

5.3.125

(a) 0.0750 (b) 0.1423

Question 5.339

126. Find the following conditional probabilities for a randomly chosen patient:

  1. Completed the treatment, given that the patient visited a family practice,
  2. Completed the treatment, given that the patient visited an OB/GYN practice,

Question 5.340

127. Based on your answers to the three previous exercises, state whether the following are independent:

  1. Visiting a family practice and completing the treatment
  2. Visiting an OB/GYN practice and completing the treatment

5.3.127

(a) Not independent (b) Not independent

WORKING WITH LARGE DATA SETS

image Chapter 5 Case Study: The Gardasil Vaccine. Open the data set Gardasil. We shall explore some probabilities about the patient completion of the vaccination regime and the age group of the patient (“0” = 11–17 years old; “1” = 18–26 years old), using the tools and techniques we have learned in this section. Use technology to do Exercises 128–131. Define the following events:

  • : Completed treatment
  • : Patient is 11–17 years old
  • : Patient is 18–26 years old

Question 5.341

gardasil

128. Construct a contingency table (crosstabulation) of age group versus whether or not they completed the treatment.

Question 5.342

gardasil

129. Use the contingency table to calculate the following probabilities:

  1. Completed the treatment
  2. Patient is 11–17 years old.
  3. Patient is 18–26 years old.
  4. Completed the treatment, and the patient is 11–17 years old
  5. Completed the treatment, and the patient is 18–26 years old
  6. Completed the treatment, given that patient is 11–17 years old
  7. Completed the treatment, given that patient is 18–26 years old

5.3.129

(a) 0.3319 (b) 0.4961 (c) 0.5039 (d) 0.1748 (e) 0.1571 (f) 0.3524 (g) 0.3118

Question 5.343

gardasil

130. Based on your work in the previous exercise, determine whether the following events are independent:

  1. Completing the treatment, and the patient being 11–17 years old
  2. Completing the treatment, and the patient being 18–26 years old.

Question 5.344

gardasil

131. Which age group had a higher proportion finishing the treatment? Do you think this difference reflects a true difference in the age groups for all patients, or is it simply the luck of the draw in this sample of patients?

5.3.131

11-17 years old; true variation