Collaborative Statistics by Barbara Illowsky, Ph.D. and Susan Dean - HTML preview

PLEASE NOTE: This is an HTML preview only and some elements such as links or page numbers may be incorrect.
Download the book in PDF, ePub, Kindle for a complete version.

CHAPTER 3. PROBABILITY TOPICS

Exercise 3.11.9

(Solution on p. 162.)

E and F mutually exclusive events. P (E) = 0.4; P (F) = 0.5. Find P (E | F).

Exercise 3.11.10

J and K are independent events. P(J | K) = 0.3. Find P (J) .

Exercise 3.11.11

(Solution on p. 162.)

U and V are mutually exclusive events. P (U) = 0.26; P (V) = 0.37. Find:

a. P(U AND V) =

b. P(U | V) =

c. P(U OR V) =

Exercise 3.11.12

Q and R are independent events. P (Q) = 0.4 ; P (Q AND R) = 0.1 . Find P (R).

Exercise 3.11.13

(Solution on p. 162.)

Y and Z are independent events.

a. Rewrite the basic Addition Rule P(Y OR Z)

= P (Y) + P (Z) − P (Y AND Z) using the

information that Y and Z are independent events.

b. Use the rewritten rule to find P (Z) if P (Y OR Z) = 0.71 and P (Y) = 0.42 .

Exercise 3.11.14

G and H are mutually exclusive events. P (G) = 0.5; P (H) = 0.3

a. Explain why the following statement MUST be false: P (H | G) = 0.4 .

b. Find: P(H OR G).

c. Are G and H independent or dependent events? Explain in a complete sentence.

Exercise 3.11.15

(Solution on p. 162.)

The following are real data from Santa Clara County, CA. As of a certain time, there had been

a total of 3059 documented cases of AIDS in the county. They were grouped into the following

categories (Source: Santa Clara County Public H.D.):

Homosexual/Bisexual

IV Drug User*

Heterosexual Contact

Other

Totals

Female

0

70

136

49

____

Male

2146

463

60

135

____

Totals

____

____

____

____

____

Table 3.5: * includes homosexual/bisexual IV drug users

Suppose one of the persons with AIDS in Santa Clara County is randomly selected. Compute the

following:

a. P(person is female) =

b. P(person has a risk factor Heterosexual Contact) =

c. P(person is female OR has a risk factor of IV Drug User) =

d. P(person is female AND has a risk factor of Homosexual/Bisexual) =

e. P(person is male AND has a risk factor of IV Drug User) =

f. P(female GIVEN person got the disease from heterosexual contact) =

g. Construct a Venn Diagram. Make one group females and the other group heterosexual contact.

Available for free at Connexions <http://cnx.org/content/col10522/1.40>

147

Exercise 3.11.16

Solve these questions using probability rules. Do NOT use the contingency table above. 3059

cases of AIDS had been reported in Santa Clara County, CA, through a certain date. Those cases

will be our population. Of those cases, 6.4% obtained the disease through heterosexual contact

and 7.4% are female. Out of the females with the disease, 53.3% got the disease from heterosexual

contact.

a. P(person is female) =

b. P(person obtained the disease through heterosexual contact) =

c. P(female GIVEN person got the disease from heterosexual contact) =

d. Construct a Venn Diagram. Make one group females and the other group heterosexual contact.

Fill in all values as probabilities.

Exercise 3.11.17

(Solution on p. 163.)

The following table identifies a group of children by one of four hair colors, and by type of hair.

Hair Type

Brown

Blond

Black

Red

Totals

Wavy

20

15

3

43

Straight

80

15

12

Totals

20

215

Table 3.6

a. Complete the table above.

b. What is the probability that a randomly selected child will have wavy hair?

c. What is the probability that a randomly selected child will have either brown or blond hair?

d. What is the probability that a randomly selected child will have wavy brown hair?

e. What is the probability that a randomly selected child will have red hair, given that he has

straight hair?

f. If B is the event of a child having brown hair, find the probability of the complement of B.

g. In words, what does the complement of B represent?

Exercise 3.11.18

A previous year, the weights of the members of the San Francisco 49ers and the Dallas Cowboys

were published in the San Jose Mercury News. The factual data are compiled into the following

table.

Shirt#

210

211-250

251-290

290

1-33

21

5

0

0

34-66

6

18

7

4

66-99

6

12

22

5

Table 3.7

For the following, suppose that you randomly select one player from the 49ers or Cowboys.

a. Find the probability that his shirt number is from 1 to 33.

b. Find the probability that he weighs at most 210 pounds.

Available for free at Connexions <http://cnx.org/content/col10522/1.40>

148

CHAPTER 3. PROBABILITY TOPICS

c. Find the probability that his shirt number is from 1 to 33 AND he weighs at most 210 pounds.

d. Find the probability that his shirt number is from 1 to 33 OR he weighs at most 210 pounds.

e. Find the probability that his shirt number is from 1 to 33 GIVEN that he weighs at most 210

pounds.

f. If having a shirt number from 1 to 33 and weighing at most 210 pounds were independent

events, then what should be true about P(Shirt# 1-33 | ≤ 210 pounds)?

Exercise 3.11.19

(Solution on p. 163.)

Approximately 281,000,000 people over age 5 live in the United States.

Of

these

people,

55,000,000

speak

a

language

other

than

English

at

home.

Of

those

who

speak

another

language

at

home,

62.3%

speak

Spanish.

(Source:

http://www.census.gov/hhes/socdemo/language/data/acs/ACS-12.pdf )

Let: E = speak English at home; E’ = speak another language at home; S = speak Spanish;

Finish each probability statement by matching the correct answer.

Probability Statements

Answers

a. P(E’) =

i. 0.8043

b. P(E) =

ii. 0.623

c. P(S and E’) =

iii. 0.1957

d. P(S|E’) =

iv. 0.1219

Table 3.8

Exercise 3.11.20

The probability that a male develops some form of cancer in his lifetime is 0.4567 (Source: Ameri-

can Cancer Society). The probability that a male has at least one false positive test result (meaning

the test comes back for cancer when the man does not have it) is 0.51 (Source: USA Today). Some of

the questions below do not have enough information for you to answer them. Write “not enough

information” for those answers.

Let: C = a man develops cancer in his lifetime; P = man has at least one false positive

a. Construct a tree diagram of the situation.

b. P (C) =

c. P(P|C) =

d. P(P|C’ ) =

e. If a test comes up positive, based upon numerical values, can you assume that man has cancer?

Justify numerically and explain why or why not.

Exercise 3.11.21

(Solution on p. 163.)

In 1994, the U.S. government held a lottery to issue 55,000 Green Cards (permits for non-citizens

to work legally in the U.S.). Renate Deutsch, from Germany, was one of approximately 6.5 million

people who entered this lottery. Let G = won Green Card.

a. What was Renate’s chance of winning a Green Card? Write your answer as a probability state-

ment.

b. In the summer of 1994, Renate received a letter stating she was one of 110,000 finalists chosen.

Once the finalists were chosen, assuming that each finalist had an equal chance to win, what

was Renate’s chance of winning a Green Card? Let F = was a finalist. Write your answer as

a conditional probability statement.

Available for free at Connexions <http://cnx.org/content/col10522/1.40>

149

c. Are G and F independent or dependent events? Justify your answer numerically and also

explain why.

d. Are G and F mutually exclusive events? Justify your answer numerically and also explain why.

NOTE: P.S. Amazingly, on 2/1/95, Renate learned that she would receive her Green Card – true

story!

Exercise 3.11.22

Three professors at George Washington University did an experiment to determine if economists

are more selfish than other people. They dropped 64 stamped, addressed envelopes with $10 cash

in different classrooms on the George Washington campus. 44% were returned overall. From the

economics classes 56% of the envelopes were returned. From the business, psychology, and history

classes 31% were returned. (Source: Wall Street Journal )

Let: R = money returned; E = economics classes; O = other classes

a. Write a probability statement for the overall percent of money returned.

b. Write a probability statement for the percent of money returned out of the economics classes.

c. Write a probability statement for the percent of money returned out of the other classes.

d. Is money being returned independent of the class? Justify your answer numerically and explain

it.

e. Based upon this study, do you think that economists are more selfish than other people? Explain

why or why not. Include numbers to justify your answer.

Exercise 3.11.23

(Solution on p. 163.)

The chart below gives the number of suicides estimated in the U.S. for a recent year by age, race

(black and white), and sex. We are interested in possible relationships between age, race, and sex.

We will let suicide victims be our population. (Source: The National Center for Health Statistics,

U.S. Dept. of Health and Human Services)

Race and Sex

1 - 14

15 - 24

25 - 64

over 64

TOTALS

white, male

210

3360

13,610

22,050

white, female

80

580

3380

4930

black, male

10

460

1060

1670

black, female

0

40

270

330

all others

TOTALS

310

4650

18,780

29,760

Table 3.9

NOTE: Do not include "all others" for parts (f), (g), and (i).

a. Fill in the column for the suicides for individuals over age 64.

b. Fill in the row for all other races.

c. Find the probability that a randomly selected individual was a white male.

d. Find the probability that a randomly selected individual was a black female.

e. Find the probability that a randomly selected individual was black

Available for free at Connexions <http://cnx.org/content/col10522/1.40>

150

CHAPTER 3. PROBABILITY TOPICS

f. Find the probability that a randomly selected individual was male.

g. Out of the individuals over age 64, find the probability that a randomly selected individual was

a black or white male.

h. Comparing “Race and Sex” to “Age,” which two groups are mutually exclusive? How do you

know?

i. Are being male and committing suicide over age 64 independent events? How do you know?

The next two questions refer to the following: The percent of licensed U.S. drivers (from a recent year)

that are female is 48.60. Of the females, 5.03% are age 19 and under; 81.36% are age 20 - 64; 13.61% are age

65 or over. Of the licensed U.S. male drivers, 5.04% are age 19 and under; 81.43% are age 20 - 64; 13.53% are

age 65 or over. (Source: Federal Highway Administration, U.S. Dept. of Transportation)

Exercise 3.11.24

Complete the following:

a. Construct a table or a tree diagram of the situation.

b. P(driver is female) =

c. P(driver is age 65 or over | driver is female) =

d. P(driver is age 65 or over AND female) =

e. In words, explain the difference between the probabilities in part (c) and part (d).

f. P(driver is age 65 or over) =

g. Are being age 65 or over and being female mutually exclusive events? How do you know

Exercise 3.11.25

(Solution on p. 163.)

Suppose that 10,000 U.S. licensed drivers are randomly selected.

a. How many would you expect to be male?

b. Using the table or tree diagram from the previous exercise, construct a contingency table of

gender versus age group.

c. Using the contingency table, find the probability that out of the age 20 - 64 group, a randomly

selected driver is female.

Exercise 3.11.26

Approximately 86.5% of Americans commute to work by car, truck or van. Out of that group,

84.6% drive alone and 15.4% drive in a carpool. Approximately 3.9% walk to work and approxi-

mately 5.3% take public transportation. (Source: Bureau of the Census, U.S. Dept. of Commerce.

Disregard rounding approximations.)

a. Construct a table or a tree diagram of the situation. Include a branch for all other modes of

transportation to work.

b. Assuming that the walkers walk alone, what percent of all commuters travel alone to work?

c. Suppose that 1000 workers are randomly selected. How many would you expect to travel alone

to work?

d. Suppose that 1000 workers are randomly selected. How many would you expect to drive in a

carpool?

Exercise 3.11.27

Explain what is wrong with the following statements. Use complete sentences.

a. If there’s a 60% chance of rain on Saturday and a 70% chance of rain on Sunday, then there’s a

130% chance of rain over the weekend.

b. The probability that a baseball player hits a home run is greater than the probability that he

gets a successful hit.

Available for free at Connexions <http://cnx.org/content/col10522/1.40>

index-161_1.png

151

3.11.1 Try these multiple choice questions.

The next two questions refer to the following probability tree diagram which shows tossing an unfair

coin FOLLOWED BY drawing one bead from a cup containing 3 red (R), 4 yellow (Y) and 5 blue (B) beads.

For the coin, P (H) = 2 and P (T) = 1 where H = "heads" and T = "tails”.

3

3

Figure 3.3

Exercise 3.11.28

(Solution on p. 164.)

Find P(tossing a Head on the coin AND a Red bead)

A. 23

B. 5

15

C. 6

36

D. 5

36

Exercise 3.11.29

(Solution on p. 164.)

Find P(Blue bead).

A. 15

36

B. 10

36

C. 10

12

D. 6

36

The next three questions refer to the following table of data obtained from www.baseball-almanac.com12

showing hit information for 4 well known baseball players. Suppose that one hit from the table is randomly

selected.

12http://cnx.org/content/m16836/latest/www.baseball-almanac.com

Available for free at Connexions <http://cnx.org/content/col10522/1.40>

152

CHAPTER 3. PROBABILITY TOPICS

NAME

Single

Double

Triple

Home Run

TOTAL HITS

Babe Ruth

1517

506

136

714

2873

Jackie Robinson

1054

273

54

137

1518

Ty Cobb

3603

174

295

114

4189

Hank Aaron

2294

624

98

755

3771

TOTAL

8471

1577

583

1720

12351

Table 3.10

Exercise 3.11.30

(Solution on p. 164.)

Find P(hit was made by Babe Ruth).

A. 1518

2873

B. 2873

12351

C.

583

12351

D. 4189

12351

Exercise 3.11.31

(Solution on p. 164.)

Find P(hit was made by Ty Cobb | The hit was a Home Run)

A. 4189

12351

B. 114

1720

C. 1720

4189

D.

114

12351

Exercise 3.11.32

(Solution on p. 164.)

Are the hit being made by Hank Aaron and the hit being a double independent events?

A. Yes, because P(hit by Hank Aaron | hit is a double) = P(hit by Hank Aaron)

B. No, because P(hit by Hank Aaron | hit is a double) = P(hit is a double)

C. No, because P(hit is by Hank Aaron | hit is a double) = P(hit by Hank Aaron)

D. Yes, because P(hit is by Hank Aaron | hit is a double) = P(hit is a double)

Exercise 3.11.33

(Solution on p. 164.)

Given events G and H: P(G) = 0.43 ; P(H) = 0.26 ; P(H and G) = 0.14

A. Find P(H or G)

B. Find the probability of the complement of event (H and G)

C. Find the probability of the complement of event (H or G)

Exercise 3.11.34

(Solution on p. 164.)

Given events J and K: P(J) = 0.18 ; P(K) = 0.37 ; P(J or K) = 0.45

A. Find P(J and K)

B. Find the probability of the complement of event (J and K)

C. Find the probability of the complement of event (J or K)

Available for free at Connexions <http://cnx.org/content/col10522/1.40>

153

Exercise 3.11.35

(Solution on p. 164.)

United Blood Services is a blood bank that serves more than 500 hospitals in 18 states. Accord-

ing to their website, http://www.unitedbloodservices.org/humanbloodtypes.html, a person with

type O blood and a negative Rh factor (Rh−) can donate blood to any person with any bloodtype.

Their data show that 43% of people have type O blood and 15% of people have Rh− factor; 52%

of people have type O or Rh− factor.

A. Find the probability that a person has both type O blood and the Rh− factor

B. Find the probability that a person does NOT have both type O blood and the Rh− factor.

Exercise 3.11.36

(Solution on p. 164.)

At a college, 72% of courses have final exams and 46% of courses require research papers. Suppose

that 32% of courses have a research paper and a final exam. Let F be the event that a course has a

final exam. Let R be the event that a course requires a research paper.

A. Find the probability that a course has a final exam or a research project.

B. Find the probability that a course has NEITHER of these two requirements.

Exercise 3.11.37

(Solution on p. 164.)

In a box of assorted cookies, 36% contain chocolate and 12% contain nuts. Of those, 8% contain

both chocolate and nuts. Sean is allergic to both chocolate and nuts.

A. Find the probability that a cookie contains chocolate or nuts (he can’t eat it).

B. Find the probability that a cookie does not contain chocolate or nuts (he can eat it).

Exercise 3.11.38

(Solution on p. 164.)

A college finds that 10% of students have taken a distance learning class and that 40% of students

are part time students. Of the part time students, 20% have taken a distance learning class. Let D

= event that a student takes a distance learning class and E = event that a student is a part time

student

A. Find P(D and E)

B. Find P(E | D)