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Chapter 4 - Introduction to Probability
98. The results of a survey of 800 married couples and the number of children they had is shown below.
Number
of Children
Probability
0
0.050
1
0.125
2
0.600
3
0.150
4
0.050
5
0.025
If a couple is selected at random, what is the probability that the couple will have
a.
Less than 4 children?
b.
More than 2 children?
c.
Either 2 or 3 children?
99. 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.
100. 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?
Chapter 4 - Introduction to Probability
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?
101. 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?
102. 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?
103. 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).
104. 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.
Chapter 4 - Introduction to Probability
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.
105. 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?
106. 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.
Chapter 4 - Introduction to Probability
107. 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?
108. 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?
109. A small town has 5,600 residents. The residents in the town were asked whether or not they favored building a new
Chapter 4 - Introduction to Probability
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.
110. 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.
Chapter 4 - Introduction to Probability
111. 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?
112. 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?
113. 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?
114. 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.
115. 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?
116. 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?
117. 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?
118. 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.
119. The probability of an economic decline in the year 2013 is 0.23. There is a probability of 0.64 that we will elect a
republican president in the year 2012. 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 2012 and an economic decline in
the year 2013?
c.
If we experience an economic decline in the year 2013, what is the probability that a
Republican president will have been elected in the year 2012?
d.
What is the probability of economic decline in 2013 or a Republican president elected in the
year 2012 or both?
120. 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.
121. 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
Chapter 4 - Introduction to Probability
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?
122. 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?
123. 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.
124. 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.
125. 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.
Chapter 4 - Introduction to Probability
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?
126. 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?
127. 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.
128. 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.
129. 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(A∩C), P(B∩C).
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?
130. 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.
131. 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?
132. 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?
133. 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?
Chapter 4 - Introduction to Probability
c. Given that a driver has no accident, what is the probability the driver is a female?
134. 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?
135. 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?
136. The following probability model describes the number of snowstorms for Washington, D.C. for a given year:
Number of
Snowstorms
0
1
2
3
4
5
6
Probability
.25
.33
.24
.11
.04
.02
.01
The probability of 7 or more snowstorms in a year is 0.
a. What is the probability of more than 2 but less than 5 snowstorms?
b. Given this a particularly cold year (in which 2 snowstorms have already been observed), what is the conditional
probability that 4 or more snowstorms will be observed?
Chapter 4 - Introduction to Probability
c. If at the beginning of winter there is a snowfall, what is the probability of at least one more snowstorm before winter is
over?
137. The Ambell Company uses batteries from two different manufacturers. Historically, 60% of the batteries are from
manufacturer 1, and 90% of these batteries last for over 40 hours. Only 75% of the batteries from manufacturer 2 last for
over 40 hours. A battery in a critical tool fails at 32 hours. What is the probability it was from manufacturer 2?
138. It is estimated that 3% of the athletes competing in a large tournament are users of an illegal drug to enhance
performance. The test for this drug is 90% accurate. What is the probability that an athlete who tests positive is actually a
user?
139. Through a telephone survey, a low-interest bank credit card is offered to 400 households. The responses are as
follows.
Income < $50,000
Income > $50,000
Accept offer
40
30
Reject offer
210
120
a. Develop a joint probability table and show the marginal probabilities..
b. What is the probability of a household whose income exceeds $50,000 and who rejects the offer?
c. If income is < $50,000, what is the probability the offer will be accepted?
d. If the offer is accepted, what is the probability that income exceeds $50,000?
140. There are two more assignments in a class before its end, and if you get an A on at least one of them, you will get an
A for the semester. Your subjective assessment of your performance is
Event
Probability
A on paper and A on exam
.25
A on paper only
.10
A on exam only
.30
Chapter 4 - Introduction to Probability
A on neither
.35
a. What is the probability of getting an A on the paper?
b. What is the probability of getting an A on the exam?
c. What is the probability of getting an A in the course?
d. Are the grades on the assignments independent?
141. A marina has two party boats available for customers to rent. Historically, demand for party boats has followed this
distribution shown below. The revenue per rental is $400. If a customer wants a party boat and none is available, the
store gives a $150 coupon for jet ski rental.
Demand
Relative Frequency
Revenue
Cost
0
.35
0
0
1
.30
400
0
2
.20
800
0
3
.10
800
150
4
.05
800
300
a. What is the expected demand?
b. What is the expected revenue?
c. What is the expected cost?
d. What is the expected profit?
142. To better track its patients, a hospital's neighborhood medical center has gathered this information.
New patient (N)
Existing patient (E)
Scheduled appointment (A)
10
10
Walk-in (W)
12
18
a.
Develop a joint probability table. Include the marginal probabilities.
b.
Find the conditional probabilities:
P(A|N), P(A|E), P(W|N), P(W|E), P(N|A), P(E|A), P(N|W), P(E|W)
143. A medical research project examined the relationship between a subject's weight and recovery time from a surgical
procedure, as shown in the table below.
Underweight
Normal weight
Overweight
Less than 3 days
6
15
3
3 to 7 days
30
95
20
Over 7 days
14
40
27
a.
Use relative frequency to develop a joint probability table to show the marginal probabilities.
b.
What is the probability a patient will recover in fewer than 3 days?
c.
Given that recovery takes over 7 days, what is the probability the patient is overweight?
144. A mail order company tracks the number of returns it receives each day. Information for the last 50 days shows
Number of returns
Number of days
0 - 99
6
100 - 199
20
200 - 299
15
300 or more
9
a.
How many sample points are there?
b.
List and assign probabilities to sample points.
c.
What procedure was used to assign these probabilities?
145. A package of candy contains 12 brown, 5 red, and 8 green candies. You grab three pieces from the package. Give the
sample space of colors you could get. Order is not important.
146. Mini Car Motors offers its luxury car in three colors: gold, silver and blue. The vice president of advertising is
interested in the order of popularity of the color choices by customers during the first month of sales.
a.
How many sample points are there in this experiment?
b.
If the event A = gold is the most popular color, list the outcome(s) in event A.
c.
If the event B = blue is the least popular color, list the outcome(s) in A∩ B.
d.
List the outcome(s) in A∩ Bc.
147. Providence Land Development Company has just hired four new salespersons. After six months on the job each
salesperson will be rated as either poor, average or excellent and will be compensated accordingly. Assume that
Providence is concerned with the number of salespersons in each category.
a) List the outcomes of this experiment.
b) Let the event A = at least two salespersons are rated average and let event B = exactly one salesperson is rated poor.
List the outcomes in AB.
c) Let the event C = exactly two salespersons are rated excellent. List the outcomes in D = BC.
d) Are events D and A mutually exclusive?
e) Let the event E = at most one salesperson is rated average. Are events A and E mutually exclusive?
148. Sales of the first 500 luxury Mini Cars were as follows: 250 gold, 150 silver, and 100 blue. Assume the relative
frequency method is used to assign probabilities for color choice and the color of each car sold is independent of that of
any other car sold.
a) What is the probability that the next two cars sold will be gold?
b) What is the probability that neither of the next two cars sold will be silver?
c) What is the probability that of the next two cars sold, one will be silver and the other will be blue?
Chapter 4 - Introduction to Probability
149. The sales manager for Widco Distributing Company has estimated demand for a new 15-piece cutlery set will be
between 0 and 2 units per day. He believes the probability of selling no sets is .65, of one set is .25, and two sets is .10.
The company is interested in sales over a two day period, assuming sales from day to day are independent.
a) What is the probability of selling no sets during the two days?
b) What is the probability of selling one set during the two days?
c) What is the probability of selling three or more sets during the two days?
d) What is the probability of selling twosets during the two days?
150. Higbee Manufacturing Corp. has recently received 5 cases of a certain part from one of its suppliers. The defect rate
for the parts is normally 5%, but the supplier has just notified Higbee that one of the cases shipped to them has been made
on a misaligned machine that has a defect rate of 97%. So the plant manager selects a case at random and tests a part.
a) What is the probability that the part is defective?
b) Suppose the part is defective, what is the probability that this is from the case made on the misaligned machine?
c) After finding that the first part was defective, suppose a second part from the case is tested. However, this part is
found to be good. Using the revised probabilities from part (b) compute the new probability of these parts being from the
defective case.
