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EQT 272
PROBABILITY
AND STATISTICS
ROHANA BINTI ABDUL HAMID
INSTITUT E FOR ENGINEERING MATHEMATICS (IMK)
UNIVERSITI MALAYSIA PERLIS
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CHAPTER 1
PROBABILITY
1.1 Introduction
1.2 Sample space and algebra of sets
1.3 Tree diagrams and counting techniques
1.4 Properties of probability
1.5 Conditional probability
1.6 Bayes’s theorem
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1.7 Independence
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WHY DO COMPUTER ENGINEERS NEED
TO STUDY PROBABILITY???????
1.
2.
3.
4.
5.
Signal processing
Computer memories
Optical communication systems
Wireless communication systems
Computer network traffic
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Probability and statistics are related
in an important way.
Probability is used as a tool; it allows
you to evaluate the reliability of your
conclusions about the population when
you have only sample information.
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Probability
• Probability is a measure of the likelihood
of an event A occurring in one experiment
or trial and it is denoted by P (A).
number of ways thattheevent A can occur ( A)
P( A) 
totalnumber of outcomes( S )
n( A)

n( S )
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Experiment
• An experiment is any process of
making an observation leading to
outcomes for a sample space.
Example:
-Toss a die and observe the number that
appears on the upper face.
-A medical technician records a person’s blood
type.
-Recording a test grade.
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The mathematical basis of probability is the
theory of sets.
• Sets
A set is a collection of elements or components
• Sample Spaces, S
A sample space consists of points that
correspond to all possible outcomes.
• Events
An event is a set of outcomes of an experiment
and a subset of the sample space.
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• Experiment:
Tossing a die
• Sample space:
S ={1, 2, 3, 4, 5, 6}
• Events:
A: Observe an odd number
B: Observe a number less than 4
C: Observe a number which could
divide by 3
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Basic Operations
S
B
A
Figure 1.1: Venn diagram representation of
events
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1. The union of events A and B, which is denoted as A  B ,
- is the set of all elements that belong to A or B or both.
- Two or more events are called collective exhaustive events if the
unions of these events result in the sample space.
2. The intersection of events A and B, which is denoted by A  B,
- is the set of all elements that belong to both A and B.
- When A and B have no outcomes in common, they are said to
be mutually exclusive or disjoint sets.
3. The event that contains all of the elements that do not belong to
Freecomplement
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an event A is called the
of A and is denoted byPage
A 10
Exercise 1.1
• Given the following sets;
A= {2, 4, 6, 8, 10}
B= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
C= {1, 3, 5, 11,….}, the set of odd numbers
Find A  B , A  B and C
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Answer
• A  B = {1, 2, 3, 4, 5, 6, 7, 8, 9,10}
• A  B = {2, 4, 6, 8, 10}
• C = {2, 4, 6, 8,…}, the set of even
numbers
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1.3.1 Tree diagrams
• Some experiments can be generated in
stages, and the sample space can be
displayed in a tree diagram.
• Each successive level of branching on the
tree corresponds to a step required to
generate the final outcome.
• A tree diagram helps to find simple
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events.
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• A box contains one white and two blue
balls. Two balls are randomly selected and
their colors recorded. Construct a tree
diagram for this experiment and state the
simple events.
W1
B1
B2
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First ball
Second ball
B1
W1
B2
W1
B1
B2
RESULTS
W1B1
W1B2
B1W1
B1B2
W1
B2W1
B1
B2B1
B2
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Exercise 1.2
• 3 people are randomly selected from voter
registration and driving records to report
for jury duty. The gender of each person is
noted by the county clerk. List the simple
events by creating a tree diagram.
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1.3.2 Counting technique
• We can use counting techniques or counting
rules to
# find the number of ways to accomplish the
experiment
# find the number of simple events.
# find the number of outcomes
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Permutations
Counting
rules
Combinations
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• This counting rule count the
number of outcomes when the
experiment involves selecting r
objects from a set of n objects
when the order of selection is
important.
n
n!
Pr 
( n  r )!
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• The number of ways to arrange
an entire set of n distinct items is
n
Pn  n!
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• Suppose you have 3 books, A, B and C
but you have room for only two on your
bookshelf. In how many ways can you
select and arrange the two books when
the order is important.
A
B
C
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A
B
A
C
A
B
C
B
A
2.AC
C
A
3.BC
C
A
C
B
1.AB
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4.BA
5.CA
6.CB
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n
3
n!
Pr 
( n  r )!
3!
P2 
( 3 2 )!
6
There are 6 ways to select and
arrange the books
in order.
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Exercise 1.3
Three lottery tickets are drawn from a
total of 50. If the tickets will be distributed
to each of the employees in the order in
which they are drawn, the order will be
important. How many simple events are
associated with the experiment?
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• This counting rule count the
number of outcomes when the
experiment involves selecting r
objects from a set of n objects
when the order of selection is not
important.
n!
nC   n  
r r 
  r ! n  r  !
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• Suppose you have 3 books, A, B and C
but you have room for only two on your
bookshelf. In how many ways can you
select and arrange the two books when
the order is not important.
A
B
C
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A
B
1.AB
A
C
2.AC
A
B
C
3.BC
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n!
nC   n  
r r 
  r ! n  r  !
3
3!
C2 
2!( 3 2 )!
3
There are 3 ways to select and arrange
the books when the order is not
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important
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Exercise 1.4
Suppose that in the taste test, each
participant samples 8 products and is
asked the 3 best products, but not in any
particular order. Calculate the number of
possible answer test.
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1) 0  P ( A)  1
2) P ( A)  P ( A)  1
3) P ( A  B)  P ( A)  P ( A  B )
4) P ( A  B )  P ( B )  P ( A  B )
5) P ( A  B)  1  P ( A  B )
6) P (( A  B ))  P ( A  B )
7) P (( A  B ))  P ( A  B)
8) P ( A  ( A  B ))  P ( A  B )
9)
S
B
A
A B
A  B
A  B
P ( B )  P[( A  B )  ( A  B )]
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Theorem 1.1 : Laws of
Probability
a) P( A)  1 – P  A 
b) P( A  B)  P  A   P  B  – P( A  B)
c) P( A  B  C )  P  A   P  B   P  C  – P( A  B) – P( A  C ) – P( B  C )  P( A  B  C )
d) If A and B are mutually exclusive events, then P( A  B)  0
e) If A1 and A2 are the subset of S where A1  A2 , then P  A1   P  A2 
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Two fair dice are thrown. Determine
a) the sample space of the experiment
b) the elements of event A if the outcomes of both
dice thrown are showing the same digit.
c) the elements of event B if the first thrown giving
a greater digit than the second thrown.
d) probability of event A, P(A) and event B, P(B)
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Solutions 1.5
a) Sample space, S
1
2
3
4
5
6
1
(1, 1)
(1, 2)
(1, 3)
(1, 2)
(1, 5)
(1, 6)
2
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
3
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
4
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
5
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
6
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)
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Solutions 1.5
b) A = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}
c) B = {(2, 1), (3, 1), (3, 2), (4, 1), (4, 2), (4, 3), (5, 1), (5, 2), (5, 3),
(5, 4), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5)}
n( A) 6 1
d) P  A 


n( S ) 36 6
n( B) 15 5
P  B 


n( S ) 36 12
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Consider randomly selecting a UniMAP Master Degree
international student, and let A denote the event that the
selected individual has a Visa Card and B has a
Master Card. Suppose that P(A) = 0.5 and P(B) = 0.4
and P( A  B) = 0.25.
a) Compute the probability that the selected individual
has at least one of the two types of cards ?
b) What is the probability that the selected individual
has neither type of card?
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Solutions 1.6
a) P( A  B)  P  A   P  B  – P( A  B)
= 0.5  0.4 – 0.25  0.65
b) 1  P( A  B)  1 – 0.65  0.35
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• Definition:
For any two events A and B with P(B) >
0, the conditional probability of A given
that B has occurred is defined by
P( A  B)
P( A | B) 
P( B)
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A study of 100 students who get A in Mathematics in
SPM examination was done by UniMAP first year
students. The results are given in the table :
Area/Gender
Male (C)
Female (D)
Total
Urban (A)
35
10
45
Rural (B)
25
30
55
Total
60
40
100
If a student is selected at random and have been told
that the individual is a male student, what is the
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probability of he is from
urban area?
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In 2006, Edaran Automobil Negara (EON) will
produce a multipurpose national car (MPV)
equipped with either manual or automatic
transmission and the car is available in one of
four metallic colours. Relevant probabilities
for various combinations of transmission type
and colour are given in the accompanying
table:
Transmission
Black
Grey (C)
Blue
Automatic, (A)
0.15
0.10
0.10
0.10
Manual
0.15
0.05
0.15
0.20
type/Colour
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(B)
Red
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• Let,
A = automatic transmission
B = black
C = grey
Calculate;
a) P ( A), P ( B ) and P ( A  B )
b) P ( A | B ) and P ( B | A)
c) P ( A | C ) and P ( A | C  )
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If A1 , A2 ,..., An is a partition of a sample space, then the posterior
probabilities of events Ai conditional on an event B can be obtained
from the probabilities P  Ai  and P  B | Ai  using the formula,
P  Ai  B  P  Ai  P  B | Ai 
P  Ai | B  


P  B
P B
P  Ai  P  B | Ai 
 P A  PB | A 
n
j 1
j
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j
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There are three boxes: Box 1 contains one red ball and
three white balls; box 2 contains two red balls and
two white balls; box 3 contains three red balls and
one white ball. A box is selected at random and then
a ball is chosen at random from the selected box.
Determine the conditional probability that box 1 was
selected, given that red ball is chosen.
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• Definition :
 Two events A and B are said to be independent
if and only if either
P ( A | B )  P ( A)
or
P ( B | A)  P ( B )
Otherwise, the events are said to be dependent.
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Multiplicative Rule of Probability:
The probability that both two events A and B, occur is
P( A  B)  P  A  P  B | A 
 P  B P  A | B
If A and B are independent,
P( A  B)  P  A  P  B 
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3
1
Suppose that P( A)  and P( B)  . Are events A and B independent or
5
3
mutually exclusive if ,
1
a) P( A  B) 
5
14
b) P( A  B) 
15
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