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Exercises (Probability)
Stat 120
Fall 2009 – 2010
I. Multiple Choice Questions: Please circle the correct answer.
1. Two independent events are
a. always mutually exclusive
b. never mutually exclusive
c. always complementary
d. sometimes mutually exclusive
2. Two equally likely events
a. have the same probability of occurrence
c. have no effect on the occurrence of each other
b. cannot occur together
d. are dependent
3. Suppose P(A) = 0.40, P(B) = 0.20 and PA  B  0.08 . Based on these, can you
tell if A and B are
a. mutually exclusive b. independent c. both (a) & (b) d. none of the above
4. When sampling without replacement from a finite population,
a. the second Pick dependent on the first pick.
b. a complement of the first pick
c. mutually exclusive from the first pick.
d. independent from the first pick
5. Which of the following is a correct statement about a probability?
a. It may assume negative values.
b. It ranges from 0 to 1.
c. It cannot be reported to more than 1 decimal place.
d. It may be greater than 1
6. Which of the following is not an approach of calculating probability?
a. Subjective
b. Independent
c. Relative frequency d. Classical
7. According to the classical definition of probability
a. All the events are equally likely to occur.
b. The probability is based on experience and belief.
c. Divide the number of successes by the number of failures
d. One outcome is exactly twice the other.
8. If P(A) = 0.3, P(B) = 0.6, and A and B are independent events, then
a. P(A  B) = 0
b. P( A  B) = 0
c. P(A) does not equal P(A|B)
d. P(A  B) = 0.72.
9. Given P(A) = 0.4, P(B) = 0.6, P(A OR B) = 0.65; Which of the following
is False?
a. P(A OR B) = P(A) + P(B) – P(A AND B)
b. P(A) = P(B')
c. A and B are dependent events
d. A and B are mutually exclusive
10. 55 percent of students are females. 80% of the females love math, while only
60% of the males love math. What percentage of the students love math?
a. 70%
b. 50%
c. 71%
d. 60%
11. P(A) = 0.29 and P(B) = 0.17. If A and B are mutually exclusive, P(A|B) =?
a. 0.29
b. 0.17
c. 0.0493
d. 0
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12. An urn contains 5 white balls and 5 black balls. One ball is drawn at random from
the urn and then thrown away. A second ball is then drawn from the urn. The
probability that the second ball is white equals
a. 1 3
b. 4 9
c. 1 2
d. 5 9
13. A survey indicates that 30% of college students drink coffee, 72% of college
students drink tea, and 25% of them drink both. If one student is selected at random,
what is the probability that the student drinks coffee or tea?
a. 0.77
b. 0.70
c. 0.28
d. 0.75
II. True/ False questions: Choose the best answer.
1. An event and its complement are independent.
2. If events A and B are mutually exclusive, the union of events A and B is the sample
space.
3. If E1 , E 2 ,..., E n are mutually exclusive and exhaustive events, the probability of
their union must be one.
4. Suppose A and B are independent events where P(A) = 0.4 and P(B) =0.5. Then
P(A|B)=0.5
5. If P(A) = 0.4 and P(A|B) = 0.5, then there is a 20% chance that A and B
Occur at the same time.
6. Hamad and Fahd go to a coffee shop during their lunch break. Each one of
them wants to pay. They decided to toss a coin to see who will pay. The
probability that Hamad will pay 2 consecutive days is 0.25.
7. There is 50% chance of rain in a given day and a 50% chance of rain,
independently, in the following day. Therefore, there is a 50% chance of
rain on exactly one of the two days.
III. Solve the following questions (Justify your answers)
1. Suppose that P(A)=0.12, P(B)=0.16.
a. What can we say about P(A|B) if A and B are independent? P(A|B)=______
b. What can we say about P(A|B) if A and B are mutually exclusive? P(A|B)=_____
2. A multiple choice quiz has two questions, each with five answers. If you just guess
on all of the questions,
a. what is the probability that you get them all right?
b. what is the probability that you get them all wrong?
3. A test of statistics is to be given next week. Suppose that 75% of the students study
for the test and 25% do not. If a student studies for the exam, the probability that he or
she will pass is 0.90. If the student does not study for the exam, the probability that he
or she will pass is 0.20. If a student is selected randomly,
a. What is the probability that he or she will pass?
b. Given that the student passed the exam, what is the probability that he or she
studied?
4 Of a total of 100 CDs manufactured on two machines, 20 are defective. Sixty of the
total CDs were manufactured on machine I, and 10 of these 60 are defective. One CD
is selected randomly from these 100 CDs. What is the probability that this CD is
a. defective
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b. defective and manufactured on machine I
c. defective given that it was manufactured on machine I
d. Neither defective nor manufactured on machine I
5. A small manufacturing plant has sales offices located in four cities: Dallas, Seattle,
Boston, and Los Angeles. An analysis of the plant's Accounts receivables reveals the
number of overdue invoices by days, as shown here.
Days Overdue
Dallas
Seattle
Boston
Los Angeles
Under 30 days
137
122
198
287
30-60 days
85
46
76
109
61-90 days
33
27
55
48
Over 90 days
18
32
45
66
Assume the invoices are sorted and managed from a central database.
I. What is the probability that a randomly selected invoice from the database is
a. from the Boston sales office?
b. between 30 and 90 days overdue?
c. over 90 days old and from the Seattle office?
II. If a randomly selected invoice is from the Los Angeles office, what is the
probability that it is 60 or fewer days overdue?
III. Is the event "the invoice is 60 or fewer days overdue" independent of being
selected from the "Los Angeles office"? Justify.
6. A basketball team has 10 players. Five are seniors, 2 are juniors, and 3 are
sophomores. Two players are randomly selected to serve as captains for the next
game.
a. What is the probability that both players selected are seniors?
b. What is the probability that only one of the two players selected is senior?
7. A salesperson either makes a sale S or does not make a sale N with each of two
potential customers. The basic outcomes and probabilities of their occurrence are as
follows:
BASIC OUTCOME
PROBABILITY
(N, N)
(N, S)
(S, N)
(S, S)
.65
.15
.12
???
a. Use the laws of probability to find the probability of (S, S), the basic outcome
where sales are made to both customer one and customer two.
b. Find the probability that at least one sale is made.
c. Find the probability that exactly one sale is made.
8. An aerospace company has submitted bids on two separate contracts, A and B. The
company has a 50% chance of winning contract A and a 40% chance of winning
contract B. Furthermore, it believes that winning contract A is independent of winning
contract B.
a. What is the probability that the company will win both contracts?
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b. What is the probability that the company will win at least one of the two contracts?
c. Now, suppose that the aerospace company believes that it has a 60% chance of
winning contract A and a 30% chance of winning contract B. Given that it wins
contract B, the company believes that it has an 80% chance of winning contract A.
i. What is the probability that the company will win both contracts?
ii. What is the probability that the company will win at least one of the two contracts?
iii. If the company wins contract B, what is the probability that it will not win contract
A?
9. Five hundred employees were selected from a city's large private companies and
they were asked whether or not they have any retirement benefits provided by their
companies. Based on this information, the following two-way table was prepared:
Men
Women
Have Retirement
Yes
225
150
Benefits
No
75
50
a. If one employee is selected at random from these 500 employees, find the
probability that this employee
i. is a woman
ii. has retirement benefits
iii. has retirement benefits given the employee is a man
b. Are the events "men" and "yes" mutually exclusive? Why or why not?
c. Are the events "woman" and "yes" independent? Why or why not?
10. A sample space contains only four sample points: E1, E2, E3, and E4.
Suppose that PE1  PE 2 and PE3  PE 4  0.3 . Find PE 2
11. A survey of publishing jobs indicates that 92 percent are completed on time.
Assume that two jobs are selected for study.
a. What is the probability that they are all completed on time?
b. What is the probability that at least one was not completed on time?
12. There are 200 students in a particular graduate program at a state university. Of
them, 110 are female and 125 are out-of-state students. Of the 110 females, 70 are
out-of-state students.
a. If one of these 200 students is selected at random, what is the probability that the
student selected is
i. a male?
ii. an " out-of-state student given that this student is a female?
iii. a female or out-of-state student?
b. If two of these 200 students are selected at random, what is the probability that both
of them are out-of-state students?
c. Are the events "female" and "out-of-state student" independent? Are they mutually
exclusive? Explain why or why not.
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13. Suppose that a manager og a large apartment complex provides the following
probability estimates about the number of vacancies that will exist next month.
Vacancies
Probability
0
0.05
1
0.15
2
0.35
3
0.25
4
0.10
Provide the probability of each of the following events:
a. No vacancies
b. Two or fewer vacancies
14. Suppose we have a sample space with five equally likely outcomes:
E1, E2, E3, E4, E5. Let A = {E1, E2}, B = {E3, E4}, C = {E2, E3, E5}
a. Find P(A), P(B), P A  B , and PA  B 
15. A survey of magazine subscribers showed that during the past 12 months 45.8%
rented a car for business reasons, 54% rented a car for personal reasons, and 30%
rented a car for both business and personal reasons.
a. What is the probability that a subscriber rented a car during the past 12 months for
business or personal reasons?
b. What is the probability that a subscriber did not rent a car during the past 12
months for either business or personal reasons?
16. Suppose that P A1   0.40 and P A2   0.60 . It is also known that
P A1  A2   0 . Suppose PB | A1   0.20 and PB | A2   0.05 .
a. Are A1 and A2 mutually exclusive? Why?
b. Compute P A1  B and P A2  B .
c. Compute PB 
d. Compute P A1 | B and P A2 | B
17. Some men and women were surveyed for their marital status. It was found that 16
were unmarried males and 34 were married males. In the whole sample of 90 persons,
34 were unmarried. Answer the following questions.
a. Compute P(male or married)
b. Compute P(male|married)
c. Compute (male and married)
18. Machines I and II respectively produce 30% and 70% of a factory's output.
Machine I produces 3% defectives and machine II produces 4% defectives. If an item
is defective, what is the probability that it was produced by machine I?
19. A survey indicates that 70% of all students live with their parents. Two students
are selected at random.
a. What is the probability that at least one lives with parents?
b. What is the probability that neither one lives with parents?
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