Chapter 2 – Introduction to Probability
61. The following probability model describes the number of snow storms for Washington, D.C. for a given year:
Number of Storms 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?
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?
62. 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 are:
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 that the driver is a male over 25?
c. Given that a driver has no accident, what is the probability the driver is a female?
d. Does knowing the fact that a driver has had no accidents give us a great deal of information regarding the driver’s
sex?
63. 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?
Chapter 2 – Introduction to Probability
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.
64. 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.
d. Do you think you would obtain the same posterior probabilities as in part (c) if the first part was not found to be
defective but the second part was?
e. Suppose, because of other evidence, the plant manager was 80% certain this case was the one made on the
misaligned machine. How would your answer to part (b) change?
65. 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.
66. 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
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?
Chapter 2 – Introduction to Probability
c. What is the probability of getting an A in the course?
d. Are the grades on the assignments independent?
67. 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?
68. Super Cola sales breakdown as 80% regular soda and 20% diet soda. While 60% of the regular soda is purchased by
men, only 30% of the diet soda is purchased by men. If a woman purchases Super Cola, what is the probability that it is a
diet soda?
69. A food distributor carries 64 varieties of salad dressing. Appleton Markets stocks 48 of these flavors. Beacon Stores
carries 32 of them. The probability that a flavor will be carried by Appleton or Beacon is 15/16. Use a Venn diagram to
find the probability a flavor is carried by both Appleton and Beacon.
Chapter 2 – Introduction to Probability
70. Through a telephone survey, a low-interest bank credit card is offered to 400 households. The responses are as tabled.
Income ≤ $60,000 Income > $60,000
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 $60,000 and who rejects the offer?
c. If income is ≤ $60,000, what is the probability the offer will be accepted?
d. If the offer is accepted, what is the probability that income exceeds $60,000?
71. 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?
Chapter 2 – Introduction to Probability
72. 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)
73. 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?
74. 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?
Chapter 2 – Introduction to Probability
75. 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.
76. 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?
77. 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?
78. 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?
Chapter 2 – Introduction to Probability
79. You are given the following information on Events A, B, C, and D.
P(A) = .4 P(A U D) = .6 P(A ) C) = .04
P(B) = .2 P(A | B) = .3 P(A ) D) = .03
P(C) = .1
a. Compute P(D).
b. Compute P(A ) B).
c. Compute P(A C).
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.
80. 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?
81. A bank has the following data on the gender and marital status of 200 customers.
Male Female
Single 20 30
Married 100 50
Chapter 2 – Introduction to Probability
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.
82. 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.
83. 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?
84. 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?
Chapter 2 – Introduction to Probability
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?
85. 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?
86. 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
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?
Chapter 2 – Introduction to Probability
Essay
87. Compare these two descriptions of probability: 1) a measure of the degree of uncertainty associated with an event, and
2) a measure of your degree of belief that an event will happen.
88. Explain the difference between mutually exclusive and independent events. Can a pair of events be both mutually
exclusive and independent?
89. Use a tree diagram, labeled with appropriate notation, to illustrate Bayes’ theorem.
90. Discuss the problems inherent in using words such as “likely,” “possibly,” or “probably” to convey degree of belief.
91. Draw a Venn diagram and label appropriately to show events A, B, their complements, intersection, and union.
92. Describe four experiments and list the experimental outcomes associated with each one.