Additional relationships among events:
- Complement of an event A: A’ =
- Union of events A and B: (A U B) or (A or B)
- Intersection of events A and B: (A  B) or (A and B) or AB
-AU
=S
- AB =  if events A and B are mutually exclusive
- Distributive laws:
A (B U C) = AB U AC
A U (BC) = (A U B) (A U C)
- De Morgan’s Laws:
Example: Probability
Suppose there are 100 majors in the department of mechanical
engineering at De La Salle University. A student is randomly selected from
these 100 majors, then let us define events as follows:
A = Student is enrolled in a calculus course,
B = Student is enrolled in a machine design course,
Of these 100 students, suppose 30 are enrolled in the calculus course, 25
are enrolled in the machine design course, and 10 in both. Find
a)
b) A  B
c) A U B
d)
Solution:
: Students not enrolled in the calculus course
A  B : Students enrolled in both the calculus and the machine design
course
A U B : Students enrolled in at least one of two courses
: Students enrolled in neither course
A
B
20
10
15
55
Therefore:
a) = {70 students}
b)
= { 10 students }
c)
= { 20 + 10 + 15} = {45 students}
d)
= { 55 students}
Example: Probability
Twenty electric motors are pulled from an assembly line and inspected for
defects. Eleven of the motors are free of defects, eight have defects on
the exterior finish, and three have defects in their assembly and will not
run. Let A denote the set of motors having assembly defects and F the set
of having defects on their finish. Using events A and F, write a symbolic
notation for the following:
a. The set of motors having both types of defects
b. The set of motors having at least one type of defect
c. The set of motors having no defects
d. The set of motors having exactly one type of defect
Solution:
11
Venn Diagram
1
2
6
A
a. A  F = AF = 2
b. A U F = A or F = 9
c.
=
= 11 (De Morgan’s Law)
d.
=7
F
Example: Probability
4.10/185) A manufacturing company has two retail outlets. It is known
that 30% of the potential customers buy products form outlet I alone,
50% buy from outlet II alone, 10% buy from both I and II, and 10% buy
from neither. Let A denote the event that a potential customer, randomly
chosen, buys from I and B denote the event that the customer buys from
II. Find the following probabilities.
a) P(A)
b) P(A U B)
c)
d) P(AB)
e)
f) P(
g) P(
Example: Probability
Suppose that three employees are to be selected from ten to visit a new
plant. (a) In how many ways can the selection be made? (b) If two of the
ten employees are female and eight are male, what is the probability that
exactly one female gets selected among the three?
Solution:
(a) Order is not important here, we merely want to know how many
subsets of size r = 3 can be selected from n = 10 people. The
results is
(b)
We have seen that there are
employees from 10.

= 120 ways to select three
Similarly, there are
= 2 ways to select one female from
the two available and
= 28 ways to select two males
from the eight available.
If selection are made at random, the probability of selecting exactly
one female is
= 7/15
Example: Probability
Five applicants for a job are ranked according to ability, with applicant
number 1 being the best, number 2 second best, and so on. These
rankings are unknown to an employer, who simply hires two applicants at
random. What is the probability that this employer hires exactly one of
the two best applicants?
Solution:
The number of possible outcomes for the process of selecting two
applicants from five is
If one of the two best is selected, the selection can be done in
ways. The other selected applicants must come from among the three
lowest-ranking applicants, which can be done in
ways. Thus, the event of interest (hiring one of the two best applicants)
can occur in 2 x 3 = 6 ways. The probability of this event is thus 6/10 =
0.60.
Examples:
4.13/186) Hydraulic assemblies for landing gear coming from an aircraft
rework facility are inspected for defects. History shows that 8% have
defects in the shafts alone, 6% have defects in the bushing alone, and
2% have defects in both the shafts and the bushings. If a randomly
chosen assembly is to be used on an aircraft, find the probability that it
has each of these defects
a. A bushing defect
b. A shaft or bushing defect
c. Only one of the two type of defects
d. No defects in shafts or bushings
4.15/194) A commercial building is designed with two entrances, door 1
and door 2. Two customers arrive and enter the building.
a. list the elements of a sample space for this observational experiment
b. If all elements in part (a) are equally likely, find the probability that
both customers use door 1; then find the probability that both customers
use the same door.
4.17/194) Among five portable generators produced by an assembly line
in one day, there are two defectives. If two generators are selected for
sale, find the probability that both will be nondefective. (Assume that two
selected for sale are chosen so that every possible sample of size two has
the same probability of being selected).
4.19/194) A package of six light bulbs contains two defective bulbs. If
three bulbs are selected for use, find the probability that none is
defective.
4.21/194) A fleet of eight taxis is to be randomly assigned to three
airports A, B, and C, with two going to A, five to B, and one to C.
a. In how many ways can this be done?
b. What is the probability that the specific cab driven by Jones is assigned
to airport C?
Example: Conditional Probability and Independence
Suppose there are 50 students enrolled in the calculus course, 45
students enrolled in the machine design course, and 10 in both. Select
one student at random from this group of 100 students.
A = Student is enrolled in a calculus course
B = Student is enrolled in a machine design course
- The probability that the selected student is enrolled in a calculus class is
P(A) = 0.50,
- The probability that the selected student is enrolled in a machine design
class is 0.45,
- The probability that the selected student is enrolled in both the classes
is 0.10.
- The probability that the student is enrolled in at least one of two classes
is
P(A U B) = P(A) + P(B) – P (A  B) = 0.50 + 0.45 – 0.10 = 0.85
- Note that the two events A and B are not mutually exclusive because
there are 10 students enrolled in both the courses. If we just add the
probabilities of two events, then we will be counting those 10 students
twice, once in event A and once with event B. Therefore we have to
subtract it once.
- The probability that the student is enrolled in none of these classes is
Example:
Records indicate that for the parts coming out of a hydraulic repair shop
at an airplane rework facility, 20% will have a shaft defect, 10% will have
a bushing defect, and 75% will be defect-free. For an item chosen at
random from this output, find the probability of the following:
A: The item has at least one type of defect
B: The item has only a shaft defect/
Solution:
- The percentages given imply that 5% of the items have both a shaft
defect and a bushing defect. Let D1 denote the event that an item has a
shaft defect and D2 the event that it has a bushing defect. Then A = D1 U
D2 and
P(A) = P(D1 U D2) = P(D1) + P(D2) – P(D1D2)
= 0.20 + 0.10 – 0.05 = 0.25
- Another possible solution is to observe that the complement of A is the
event that an item has no defects. Thus
P(A) = 1 –
= 1 – 0.75
- To find P(B), note that the event D1 that the item is a shaft defect is the
union of the event that it has only a shaft defect (B) and the event that it
has both defect (D1D2). That is, D1 = B U D1D2 where B and D1D2 are
mutually exclusive.
- Therefore, P(D1) = P(B) + P(D1D2) or
P(B) = P(D1) – P(D1D2) = 0.20 - .05 = 0.15