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7. An experiment consists of throwing two six-sided dice and observing the number of spots

on the upper faces. Determine the probability that

a. the sum of the spots is 3.

b. each die shows four or more spots.

c. the sum of the spots is not 3.

d. neither a one nor a six appear on each die.

e. a pair of sixes appear.

f. the sum of the spots is 7.

8. Assume that in your hand you hold an ordinary six-sided die and a dime. You toss both

the die and the dime on a table.

a. What is the probability that a head appears on the dime and a six on the die?

b. What is the probability that a tail appears on the dime and any number more than

3 on the die?

c. What is the probability that a number larger than 2 appears on the die?

9. A very short quiz has one multiple-choice question with five possible choices (a, b, c, d,

e) and one true or false question. Assume you are taking the quiz but do not have any

idea what the correct answer is to either question, but you mark an answer anyway.

a. What is the probability that you have given the correct answer to both questions?

b. What is the probability that only one of the two answers is correct?

c. What is the probability that neither answer is correct?

d. What is the probability that only your answer to the multiple-choice question is

correct?

e. What is the probability that you have only answered the true or false question

correctly?

10. Two of the cylinders in an eight-cylinder car are defective and need to be replaced. If two

cylinders are selected at random, what is the probability that

a. both defective cylinders are selected?

b. no defective cylinder is selected?

c. at least one defective cylinder is selected?

11. Assume two events A and B are mutually exclusive and, furthermore, P(A) = 0.2 and

P(B) = 0.4.

a. Find P(A B).

b. Find P(A B).

c. Find P(AB).

12. You are given the following information on Events A, B, C, and D.

P(A) = .4

P(A D) = .6

P(A C) = .04

P(B) = .2

P(AB) = .3

P(A D) = .03

P(C) = .1

a. Compute P(D).

b. Compute P(A B).

c. Compute P(AC).

d. Compute the probability of the complement of C.

e. Are A and B mutually exclusive? Explain your answer.

f. Are A and B independent? Explain your answer.

g. Are A and C mutually exclusive? Explain your answer.

h. Are A and C independent? Explain your answer.

13. A government agency has 6,000 employees. The employees were asked whether they

preferred a four-day work week (10 hours per day), a five-day work week (8 hours per

day), or flexible hours. You are given information on the employees’ responses broken

down by gender.

Male

Female

Total

Four days

300

600

900

Five days

1,200

1,500

2,700

Flexible

300

2,100

2,400

Total

1,800

4,200

6,000

a. What is the probability that a randomly selected employee is a man and is in

favor of a four-day work week?

b. What is the probability that a randomly selected employee is female?

c. A randomly selected employee turns out to be female. Compute the probability

that she is in favor of flexible hours.

d. What percentage of employees is in favor of a five-day work week?

e. Given that a person is in favor of flexible time, what is the probability that the

person is female?

f. What percentage of employees is male and in favor of a five-day work week?

14. A bank has the following data on the gender and marital status of 200 customers.

Male

Female

Single

20

30

Married

100

50

a. What is the probability of finding a single female customer?

b. What is the probability of finding a married male customer?

c. If a customer is female, what is the probability that she is single?

d. What percentage of customers is male?

e. If a customer is male, what is the probability that he is married?

f. Are gender and marital status mutually exclusive?

g. Is marital status independent of gender? Explain using probabilities.

15. A survey of a sample of business students resulted in the following information regarding

the genders of the individuals and their major.

Major

Gender

Management

Marketing

Others

Total

Male

40

10

30

80

Female

30

20

70

120

Total

70

30

100

200

a. What is the probability of selecting an individual who is majoring in Marketing?

b. What is the probability of selecting an individual who is majoring in

Management, given that the person is female?

c. Given that a person is male, what is the probability that he is majoring in

Management?

d. What is the probability of selecting a male individual?

16. The following table shows the number of students in three different degree programs and

whether they are graduate or undergraduate students:

Degree Program

Undergraduate

Graduate

Total

Business

150

50

200

Engineering

150

25

175

Arts & Sciences

100

25

125

Total

400

100

500

a. What is the probability that a randomly selected student is an undergraduate?

b. What percentage of students is engineering majors?

c. If we know that a selected student is an undergraduate, what is the probability

that he or she is a business major?

d. A student is enrolled in the Arts and Sciences school. What is the probability

that the student is an undergraduate student?

e. What is the probability that a randomly selected student is a graduate Business

major?

17. A small town has 5,600 residents. The residents in the town were asked whether or not

they favored building a new bridge across the river. You are given the following

information on the residents' responses, broken down by gender.

Men

Women

Total

In Favor

1,400

280

1,680

Opposed

840

3,080

3,920

Total

2,240

3,360

5,600

Let: M be the event a resident is a man

W be the event a resident is a woman

F be the event a resident is in favor

P be the event a resident is opposed

a. Find the joint probability table.

b. Find the marginal probabilities.

c. What is the probability that a randomly selected resident is a man and is in favor

of building the bridge?

d. What is the probability that a randomly selected resident is a man?

e. What is the probability that a randomly selected resident is in favor of building

the bridge?

f. What is the probability that a randomly selected resident is a man or in favor of

building the bridge or both?

g. A randomly selected resident turns out to be male. Compute the probability that

he is in favor of building the bridge.

18. On a recent holiday evening, a sample of 500 drivers was stopped by the police. Three

hundred were under 30 years of age. A total of 250 were under the influence of alcohol.

Of the drivers under 30 years of age, 200 were under the influence of alcohol.

Let A be the event that a driver is under the influence of alcohol.

Let Y be the event that a driver is less than 30 years old.

a. Determine P(A) and P(Y).

b. What is the probability that a driver is under 30 and not under the influence of

alcohol?

c. Given that a driver is not under 30, what is the probability that he/she is under the

influence of alcohol?

d. What is the probability that a driver is under the influence of alcohol if we know

the driver is under 30?

e. Show the joint probability table.

f. Are A and Y mutually exclusive events? Explain.

g. Are A and Y independent events? Explain.

19. Six vitamin and three sugar tablets identical in appearance are in a box. One tablet is

taken at random and given to Person A. A tablet is then selected and given to Person B.

What is the probability that

a. Person A was given a vitamin tablet?

b. Person B was given a sugar tablet given that Person A was given a vitamin

tablet?

c. neither was given vitamin tablets?

d. both were given vitamin tablets?

e. exactly one person was given a vitamin tablet?

f. Person A was given a sugar tablet and Person B was given a vitamin tablet?

g. Person A was given a vitamin tablet and Person B was given a sugar tablet?

20. In a random sample of UTC students 50% indicated they are business majors, 40%

engineering majors, and 10% other majors. Of the business majors, 60% were females;

whereas, 30% of engineering majors were females. Finally, 20% of the other majors

were female.

a. What percentage of students in this sample was female?

b. Given that a person is female, what is the probability that she is an engineering

major?

21. Sixty percent of the student body at UTC is from the state of Tennessee (T), 30% percent

are from other states (O), and the remainder is international students (I). Twenty percent

of students from Tennessee live in the dormitories, whereas 50% of students from other

states live in the dormitories. Finally, 80% of the international students live in the

dormitories.

a. What percentage of UTC students lives in the dormitories?

b. Given that a student lives in the dormitory, what is the probability that she/he is

an international student?

c. Given that a student does not live in the dormitory, what is the probability that

she/he is an international student?

22. Tammy is a general contractor and has submitted two bids for two projects (A and B).

The probability of getting project A is 0.65. The probability of getting project B is 0.77.

The probability of getting at least one of the projects is 0.90.

a. What is the probability that she will get both projects?

b. Are the events of getting the two projects mutually exclusive? Explain, using

probabilities.

c. Are the two events independent? Explain, using probabilities.

23. Assume you are taking two courses this semester (A and B). Based on your opinion, you

believe the probability that you will pass course A is 0.835; the probability that you will

pass both courses is 0.276. You further believe the probability that you will pass at least

one of the courses is 0.981.

a. What is the probability that you will pass course B?

b. Is the passing of the two courses independent events? Use probability

information to justify your answer.

c. Are the events of passing the courses mutually exclusive? Explain.

d. What method of assigning probabilities did you use?

24. Assume you have applied to two different universities (let's refer to them as Universities

A and B) for your graduate work. In the past, 25% of students (with similar credentials

as yours) who applied to University A were accepted, while University B accepted 35%

of the applicants. Assume events are independent of each other.

a. What is the probability that you will be accepted in both universities?

b. What is the probability that you will be accepted to at least one graduate

program?

c. What is the probability that one and only one of the universities will accept you?

d. What is the probability that neither university will accept you?

25. Assume you have applied for two scholarships, a Merit scholarship (M) and an Athletic

scholarship (A). The probability that you receive an Athletic scholarship is 0.18. The

probability of receiving both scholarships is 0.11. The probability of getting at least one of

the scholarships is 0.3.

a. What is the probability that you will receive a Merit scholarship?

b. Are events A and M mutually exclusive? Why or why not? Explain.

c. Are the two events, A and M, independent? Explain, using probabilities.

d. What is the probability of receiving the Athletic scholarship given that you have

been awarded the Merit scholarship?

e. What is the probability of receiving the Merit scholarship given that you have

been awarded the Athletic scholarship?

26. In the two upcoming basketball games, the probability that UTC will defeat Marshall is

0.63, and the probability that UTC will defeat Furman is 0.55. The probability that UTC

will defeat both opponents is 0.3465.

a. What is the probability that UTC will defeat Furman given that they defeat

Marshall?

b. What is the probability that UTC will win at least one of the games?

c. What is the probability of UTC winning both games?

d. Are the outcomes of the games independent? Explain and substantiate your

answer.

27. The probability of an economic decline in the year 2001 is 0.23. There is a probability of

0.64 that we will elect a republican president in the year 2000. If we elect a republican

president, there is a 0.35 probability of an economic decline. Let “D” represent the event

of an economic decline, and “R” represent the event of election of a Republican

president.

a. Are “R” and “D” independent events?

b. What is the probability of electing a Republican president in 2000 and an

economic decline in the year 2001?

c. If we experience an economic decline in the year 2001, what is the probability

that a Republican president will have been elected in the year 2000?

d. What is the probability of economic decline in 2001 or a Republican president

elected in the year 2000 or both?

28. As a company manager for Claimstat Corporation there is a 0.40 probability that you will

be promoted this year. There is a 0.72 probability that you will get a promotion or a

raise. The probability of getting a promotion and a raise is 0.25.

a. If you get a promotion, what is the probability that you will also get a raise?

b. What is the probability of getting a raise?

c. Are getting a raise and being promoted independent events? Explain using

probabilities.

d. Are these two events mutually exclusive? Explain using probabilities.

29. An applicant has applied for positions at Company A and Company B. The probability

of getting an offer from Company A is 0.4, and the probability of getting an offer from

Company B is 0.3. Assuming that the two job offers are independent of each other, what

is the probability that

a. the applicant gets an offer from both companies?

b. the applicant will get at least one offer?

c. the applicant will not be given an offer from either company?

d. Company A does not offer the applicant a job, but Company B does?

30. A corporation has 15,000 employees. Sixty-two percent of the employees are male.

Twenty-three percent of the employees earn more than $30,000 a year. Eighteen percent

of the employees are male and earn more than $30,000 a year.

a. If an employee is taken at random, what is the probability that the employee is

male?

b. If an employee is taken at random, what is the probability that the employee

earns more than $30,000 a year?

c. If an employee is taken at random, what is the probability that the employee is

male and earns more than $30,000 a year?

d. If an employee is taken at random, what is the probability that the employee is

male or earns more than $30,000 a year or both?

e. The employee taken at random turns out to be male. Compute the probability

that he earns more than $30,000 a year.

f. Are being male and earning more than $30,000 a year independent?

31. A statistics professor has noted from past experience that a student who follows a

program of studying two hours for each hour in class has a probability of 0.9 of getting a

grade of C or better, while a student who does not follow a regular study program has a

probability of 0.2 of getting a C or better. It is known that 70% of the students follow the

study program. Find the probability that if a student who has earned a C or better grade,

he/she followed the program.

32. A survey of business students who had taken the Graduate Management Admission Test

(GMAT) indicated that students who have spent at least five hours studying GMAT

review guides have a probability of 0.85 of scoring above 400. Students who do not

spend at least five hours reviewing have a probability of 0.65 of scoring above 400. It

has been determined that 70% of the business students spent at least five hours reviewing

for the test.

a. Find the probability of scoring above 400.

b. Find the probability that given a student scored above 400, he/she spent at least

five hours reviewing for the test.

33. A machine is used in a production process. From past data, it is known that 97% of the

time the machine is set up correctly. Furthermore, it is known that if the machine is set

up correctly, it produces 95% acceptable (non-defective) items. However, when it is set

up incorrectly, it produces only 40% acceptable items.

a. An item from the production line is selected. What is the probability that the

selected item is non-defective?

b. Given that the selected item is non-defective, what is the probability that the

machine is set up correctly?

c. What method of assigning probabilities was used here?

34. In a recent survey in a Statistics class, it was determined that only 60% of the students

attend class on Fridays. From past data it was noted that 98% of those who went to class

on Fridays pass the course, while only 20% of those who did not go to class on Fridays

passed the course.

a. What percentage of students is expected to pass the course?

b. Given that a person passes the course, what is the probability that he/she attended

classes on Fridays?

35. Thirty-five percent of the students who enroll in a statistics course go to the statistics

laboratory on a regular basis. Past data indicates that 40% of those students who use the

lab on a regular basis make a grade of B or better. On the other hand, 10% of students

who do not go to the lab on a regular basis make a grade of B or better. If a particular

student made an A, determine the probability that she or he used the lab on a regular

basis.

36. In a city, 60% of the residents live in houses and 40% of the residents live in apartments.

Of the people who live in houses, 20% own their own business. Of the people who live

in apartments, 10% own their own business. If a person owns his or her own business,

find the probability that he or she lives in a house.

37. A market study taken at a local sporting goods store showed that of 20 people questioned,

6 owned tents, 10 owned sleeping bags, 8 owned camping stoves, 4 owned both tents and

camping stoves, and 4 owned both sleeping bags and camping stoves. Let Event A =

owns a tent, Event B = owns a sleeping bag, Event C = owns a camping stove,

and Sample Space = 20 people questioned.

a. Find P(A), P(B), P(C), P(AC), P(BC).

b. Are the events A and C mutually exclusive? Explain briefly.

c. Are the events B and C independent events? Explain briefly.

d. If a person questioned owns a tent, what is the probability he also owns a

camping stove?

e. If two people questioned own a tent, a sleeping bag, and a camping stove, how

many own only a camping stove?

f. Is it possible for 3 people to own both a tent and a sleeping bag, but not a

camping stove?

38. The Board of Directors of Bidwell Valve Company has made the following estimates for

the upcoming year's annual earnings:

P(earnings lower than this year) = .30

P(earnings about the same as this year) = .50

P(earnings higher than this year) = .20

After talking with union leaders, the human resource department has drawn the following

conclusions:

P(Union will request wage increase | lower earnings next year) = .25

P(Union will request wage increase | same earnings next year) = .40

P(Union will request wage increase | higher earnings next year) = .90

a. Calculate the probability that the company earns the same as this year and the

union requests a wage increase.

b. Calculate the probability that the company has higher earnings next year and the

union does not request a wage increase.

c. Calculate the probability that the union requests a wage increase.

39. An accounting firm has noticed that of the companies it audits, 85% show no inventory

shortages, 10% show small inventory shortages and 5% show large inventory shortages.

The firm has devised a new accounting test for which it believes the following

probabilities hold:

P(company will pass test | no shortage) = .90

P(company will pass test | small shortage)

= .50

P(company will pass test | large shortage) = .20

a. If a company being audited fails this test, what is the probability of a large or

small inventory shortage?

b. If a company being audited passes this test, what is the probability of no

inventory shortage?

40. Global Airlines operates two types of jet planes: jumbo and ordinary. On jumbo jets,

25% of the passengers are on business while on ordinary jets 30% of the passengers are

on business. Of Global's air fleet, 40% of its capacity is provided on jumbo jets. (Hint:

you have been given two conditional probabilities.)

a. What is the probability a randomly chosen business customer flying with Global

is on a jumbo jet?

b. What is the probability a randomly chosen non-business customer flying with

Global is on an ordinary jet?

41. Safety Insurance Company has compiled the following statistics. For any one-year

period:

P(accident | male driver under 25) = .22

P(accident | male driver over 25) = .15

P(accident | female driver under 25) = .16

P(accident | female driver over 25) = .14

The percentage of Safety's policyholders in each category is:

Male Under 25 20%

Male Over 25 40%

Female Under 25 10%

Female Over 25 30%

a. What is the probability that a randomly selected policyholder will have an

accident within the next year?

b. Given that a driver has an accident, what is the probability the driver is a male

over 25?

c. Given that a driver has no accident, what is the probability the driver is a female?

42. Super Cola sales breakdown as 80% regular soda and 20% diet soda. Men purchase 60%

of the regular soda, but only 30% of the diet soda. If a woman purchases Super Cola,

what is the probability that it is a diet soda?

43. An investment advisor recommends the purchase of shares in Infogenics, Inc. He has

made the following predictions:

P(Stock goes up 20% | Rise in GDP) = .6

P(Stock goes up 20% | Level GDP) = .5

P(Stock goes up 20% | Fall in GDP) = .4

An economist has predicted that the probability of a rise in the GDP is 30%, whereas the

probability of a fall in the GDP is 40%.

a. Draw a tree diagram to represent this multiple-step experiment.

b. What is the probability that the stock will go up 20%?

c. We have been informed that the stock has gone up 20%. What is the probability

of a rise or fall in the GDP?

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