chapter 1: probability theory (cont’d) -...
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Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 1
Chapter 1: Probability Theory (Cont’d)
Section 1.3: Combinations of Events
Problem (01): Consider the sample space and events in the Figure below. Calculate the
probabilities of the events:
(a) B (b) B ∩ C
(c) A ∪ C (d) A ∩ B ∩ C
(e) A ∪ B ∪ C (f) A’ ∩ B
(g) B’ ∪ C (h) A ∪ (B ∩ C)
(i) (A ∪ B) ∩ C (j) (A’ ∪ C)’
(Problem 1.3.2 in textbook)
Solution:
∩
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 2
∪
∩ ∩
∪ ∪
∩
∪
∪ ∩
∪ ∩
∪
∪ ∪
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 3
Problem (02): Let A be the event that a person is female, let B be the event that a person
has black hair, and let C be the event that a person has brown eyes. Describe
the kinds of people in the following events:
(a) A ∩ B (b) A ∪ C’
(c) A’ ∩ B ∩ C (d) A ∩ (B ∪ C)
(Problem 1.3.4 in textbook)
Solution:
∩
∪
∩ ∩
∩ ∪
Problem (03):
If P(A) = 0.50, P(A ∩ B) = 0.10, and P(A ∪ B) = 0.80, what is P(B)?
(Problem 1.3.7 in textbook)
Solution:
∪ ∩
Problem (04):
If P(A) = 0.40 and P(A ∩ B) = 0.30, what are the possible values for P(B)?
(Problem 1.3.6 in textbook)
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 4
Solution:
≤ ∪
∪ ≤
∪ ∩ ≤
≥
≤
≤
≥ ∩
∩
∩ ≤ ≤ ∪
≤ ≤
Problem (05): A car repair can be performed either on time or late and either satisfactorily
or unsatisfactorily. The probability of a repair being on time and
satisfactory is 0.26. The probability of a repair being on time is 0.74. The
probability of a repair being satisfactory is 0.41. What is the probability of
a repair being late and unsatisfactory?
(Problem 1.3.11 in textbook)
Solution:
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 5
∩
∩
∩
Problem (06): A bag contains 200 balls that are either red or blue and either dull or shiny.
There are 55 shiny red balls, 91 shiny balls, and 79 red balls. If a ball is
chosen at random: (a) What is the probability that it is either a shiny ball
or a red ball? (b) What is the probability that it is a dull blue ball?
(Problem 1.3.12 in textbook)
Solution:
∩
∩
∩
∩
∩
∩
∩
∩
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 6
∩
∪ ∩
∩ ∪ ∪
Problem (07): In a study of patients arriving at a hospital emergency room, the gender of
the patients is considered, together with whether the patients are younger
or older than 30 years of age, and whether or not the patients are admitted
to the hospital. It is found that 45% of the patients are male, 30% of the
patients are younger than 30 years of age, 15% of the patients are females
older than 30 years of age who are admitted to the hospital, and 21% of the
patients are females younger than 30 years of age. What proportion of the
patients are females older than 30 years of age who are not admitted to the
hospital?
(Problem 1.3.13 in textbook)
Solution:
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 7
Problem (08): Recall that a company’s revenue is considerably below expectation with
probability 0.08, is slightly below expectation with probability 0.19,
exactly meets expectation with probability 0.26, is slightly above
expectation with probability 0.36, and is considerably above expectation
with probability 0.11. Let A be the event that the revenue is not below
expectation. Let B be the event that the revenue is not above expectation.
(a) What is the probability of the intersection of these two events?
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 8
(b) What is the probability of the union of these two events?
(Problem 1.3.14 in textbook)
Solution:
∩
∪ ∩
Problem (09):
You are given P(A ∪ B) = 0.7 and P(A ∪ B’) = 0.9. Determine P(A)?
(Question 2: (6 points) in Midterm Exam 2007)
Solution:
∪ ∩
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 9
∩ ∪
∪ ∩
∩ ∪
∪ ∪
∪ ∪
∪ ∪
∪ ∪
Problem (10): Samples of emissions from three suppliers are classified for conformance
to air-quality specifications. The results from 100 samples are summarized
as follows:
Conforms
Yes No
1 22 8
Supplier 2 25 5
3 30 10
Let A denote the event that a sample is from supplier 1, and let B denote
the event that a sample conforms to specifications. Determine the number
of samples:
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 10
(a) A ∪ B
(b) A’ ∩ B
(c) B’
(Unknown-source problem)
Solution:
∪
∩
Problem (11): Disks of polycarbonate plastic from a supplier are analysed for scratch and
shock resistance. The results from 100 disks are summarised as follows:
Shock resistance
High Low
Scratch
resistance
High 70 9
Low 16 5
(a) If a disk is selected at random, what is the probability that its scratch
resistance is high and its shock resistance is high?
(b) If a disk is selected at random, what is the probability that its scratch
resistance is high or its shock resistance is high?
(c) Consider the event that a disk has high scratch resistance and the
event that a disk has high shock resistance. Are these two events
mutually exclusive?
(Unknown-source problem)
Solution:
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 11
∩
∪
∪ ∩
∩ ∅
∩
∩
Problem (12): Sixty percent of the students at a certain school wear neither a ring nor a
necklace. Twenty percent wear a ring and 30 percent wear a necklace. If
one of the students is chosen randomly, what is the probability that this
student is wearing:
(a) a ring or necklace
(b) a ring and a necklace
(Unknown-source problem)
Solution:
∪
∪ ∪
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 12
∪ ∩
∩ ∪
Problem (13): A small community organization consists of 20 families, of which 4 have
one child, 8 have two children, 5 have three children, 2 have 4 children,
and 1 has five children.
(a) If one of these families is chosen at random, what is the probability
it has i children, i = 1, 2, 3, 4, 5?
(b) If one of the children is randomly chosen, what is the probability this
child comes from a family having i children, i = 1, 2, 3, 4, 5?
(Unknown-source problem)
Solution:
Civil Engineering Department: Engineering Statistics (ECIV 2005)
Engr. Yasser M. Almadhoun Page 13
Problem (14): If A, B, and C form a sample space and are mutually exclusive events, is it
possible for P(A) = 0.3, P(B) = 0.4, and P(C) = 0.5? Why or why not?
(Unknown-source problem)
Solution:
∪ ∪