Chapter 1: Probability Theory (Cont’d) - site.iugaza.edu.pssite.iugaza.edu.ps/ymadhoun/files/2016/09/Lecture-02.pdf · Let A denote the event that a sample is ... shock resistance

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:

∩



∪

∩ ∩

∪ ∪

∩

∪

∪ ∩

∪ ∩

∪

∪ ∪



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)


Solution:

∩

∪

∩ ∩

∩ ∪

Problem (03):

If P(A) = 0.50, P(A ∩ B) = 0.10, and P(A ∪ B) = 0.80, what is P(B)?


Solution:

∪ ∩

Problem (04):

If P(A) = 0.40 and P(A ∩ B) = 0.30, what are the possible values for P(B)?




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?


Solution:



∩

∩

∩

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?


Solution:

∩

∩

∩

∩

∩

∩

∩

∩



∩

∪ ∩

∩ ∪ ∪

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?


Solution:



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?



(b) What is the probability of the union of these two events?


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:

∪ ∩



∩ ∪

∪ ∩

∩ ∪

∪ ∪

∪ ∪

∪ ∪

∪ ∪

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:



(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?


Solution:



∩

∪

∪ ∩

∩ ∅

∩

∩

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


Solution:

∪

∪ ∪



∪ ∩

∩ ∪

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?


Solution:



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?


Solution:

∪ ∪

Documents

Chapter 1: Probability Theory (Cont’d) - site.iugaza.edu.pssite.iugaza.edu.ps/ymadhoun/files/2016/09/Lecture-02.pdf · Let A denote the event that a sample is ... shock resistance