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STAT 319 – Probability and Statistics For Engineers
LECTURE 03
PROBABILITY
Engineering College, Hail University, Saudi Arabia
Overview
Probability is the study of random events.
The probability, or chance, that an event will happen can be
described by a number between 0 and 1:
•
A probability of 0, or 0%, means the event has no chance
of happening.
• A probability of 1/2 , or 50%, means the event is just as
likely to happen as not to happen.
• A probability of 1, or 100%, means the event is certain to
happen.
happen
•
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For instance, the probability of a coin landing heads up is
½, or 50%, This means you would expect a coin to land
“heads up” half of the time.
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2.1 - Sample Space
The sample
p space
p
of a statistical experiment,
p
,
denoted by S, is the set of all possible
outcomes of that experiment.
Ex. Roll a die
O t
Outcomes:
landing
l di with
ith a 1,
1 2,
2 3,
3 4,
4 5,
5 or 6 face
f
up.
Sample Space: S ={1, 2, 3, 4, 5, 6}
Sample Space
Example 2 : A automobile consultant records fuel type and vehicle type
for a sample of vehicles
2 Fuel types: Gasoline,
Gasoline Diesel
3 Vehicle types: Truck, Car, SUV
The Sample Space:
e1
e1
e2
e3
e4
e5
e6
2
Gasoline, Truck
Gasoline, Car
Gasoline SUV
Gasoline,
Diesel, Truck
Diesel, Car
Diesel, SUV
Car
e2
e3
e4
Car
e5
e6
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Sample Space
Example 3: Suppose that three items
are selected at random from a
manufacturing process.
process Each item is
inspected and classified defective (D)
or non-defective (N). To list the
elements of the sample space, we
construct the tree diagram.
Sample Space:
S ={DDD, DDN, DND, DNN, NDD, NDN,
NND, NNN}
2.2 Events
An event is any collection (subset) of outcomes contained
in the sample
p space
p
S.
An event is simple if it consists of exactly one outcome
and compound if it consists of more than one outcome.
Example 1:
We may be interested in the event A that the outcome
when a die is tossed is divisible by 3. This will occur if the
outcome is an element of the subset A = {3,6} of the
sample space S.
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Events – Relations from the Set Theory
1-The complement of an event A with respect to S
is the subset of all elements of S that are not in A.
W denote
We
d
t the
th complement,
l
t off A by
b the
th symbol
b l
A‘.
2-The intersection of two events A and B, denoted
by the symbol A I B , is the event containing
all elements that are common to A and B
B.
AI B
Read: A and B
A
A′
Events
Two events A and B are mutually exclusive, or
disjoint, if A I B = φ
ie, if A and B have no
elements
l
t in
i common.
3- The union of the two events A and B, denoted
by the symbol AU B, is the event containing all
the elements that belong to A or B or both.
Read A or B
AU B
A
4
B
Mutually Exclusive
A
B
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Venn Diagrams
AU B
A
AI B
B
A′
Mutually Exclusive
A
A
B
Events
Example 1 : Rolling a die. S = {1, 2, 3, 4, 5, 6}
Let A = {1, 2, 3} and B = {1, 3, 5}
A U B = {1, 2,3,5}
A I B = {{1,3}
, }
A ′ = {4,5, 6}
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Events
Example 3:
In a Venn diagram we let the sample space be a rectangle and
represent events by circles drawn inside the rectangle.
EXERCISE 2.1
List the elements of each of the following sample spaces:
(a) the set of integers between 1 and 50 divisible by 8.
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(b) the set S = {x | x2 + 4x - 5 = 0};
(c) the set of outcomes when a coin is tossed until a tail or
three heads appear.
(d) the set S = {x | a; is a continent};
(e) the set. S = {x \ 2x - 4 > 0 and X < 1}.
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Solution 2.1
(a) S = {8, 16, 24, 32, 40, 48}.
(b) For x2 + 4x − 5 = (x + 5)(x − 1) = 0, the solutions are:
x = −5
5 and x = 1.
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So, the sample space S = {−5, 1}.
(c) S = {T,HT,HHT,HHH}.
(d) S = {N. America, S. America, Europe, Asia, Africa,
Australia, Antarctica}.
(e) Solving 2x − 4 ≥ 0 gives x ≥ 2. Since we must also
have x < 1, it follows that S = φ
Exercise 2 (2.14)
Let S = {0,1,2,3,4,5,6,7,8,9} and A = {0,2,4,6,8},
B = {1,3,5,7,9},
{1 3 5 7 9} C = {2,3,4,5},
{2 3 4 5} and D = {1,6,
{1 6 7},
7}
List the elements of the sets corresponding to the
following events:
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Solution 2.2
2.4 - Probability of an Event
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2.4 - Probability of an Event
The probability of an event A corresponds to the occurrence of that
event; It is characterized by:
If the events A1, A2, A3, ……are mutually exclusive events, then :
Example: A coin is tossed twice (2 times).
What is the probability that at least one head occurs?
Solution: The sample space; for this experiment is:
If the coin is balanced, each of these outcomes would be equally likely to
occur.
Therefore, we assign a probability of w to each sample point.
Then 4w = 1,
1 or w =1/4
1/4. If A represents the event of at least one1 head
occurring, then
A = {HH, HT, TH}
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and
P ( A) =
1 1 1 3
+ + =
4 4 4 4
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Example: let A be the event that an even number turns
up and let B be the event, that, a number divisible by 3
occurs. Find P(A U B) and P( A I B )
Solution: For the events A = {2,4,6} and B = {3,6} we have
By assigning a probability of 1/9 to each odd number and 2/9 to each
even number, we have
P( A ∪ B ) =
2 1 2 2 7
+ + + =
9 9 9 9 9
P( A ∩ B ) =
2
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Probability of an Event
Theorem:
If an experiment can result in any one of N different
equally likely outcomes, and if exactly n of these outcomes
correspond to event A, then the probability of event A is
P ( A) =
10
n
N
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Example: A statistics class for engineers consists of 25 industrial,
10 mechanical, 10 electrical, and 8 civil engineering students. If a person
is randomly selected by the instructor to answer a question, find the
probability that the student chosen is (a) an industrial engineering major,
(b) a civil engineering or an electrical engineering major
major.
Solution: Denote by : M, E, and C the students majoring in industrial,
mechanical, electrical, and civil engineering, respectively. The total
number of students in the class is 53. all of which arc equally likely to be
selected.
Since 25 of the 53 students are majoring in industrial engineering, the
probability of event:
P (I ) =
25
53
Since 18 of the 53 students are civil or electrical engineering majors, it
follows that:
P (C U E ) =
18
28
53
Additive Rules
Theorem 1: If A and B are two events, then
P( A U B ) = P ( A) + P ( B ) − P ( A ∩ B )
Corollary 1: If A and B are mutually exclusive, then
P( A U B ) = P ( A) + P ( B )
If A and B are mutually exclusive, then
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P ( A I B ) = 0.
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Additive Rules
Corollary 2: If A1, A2, ……..An are mutually exclusive, then
P( A1 U A2 ∪ ....... An ) = P ( A1 ) + P ( A2 )........
)
+ P ( An )
Theorem 2: If A and A’ are complementary events, then
P( A) + P ( A' ) = 1
Example: A card is drawn from a well-shuffled deck of 52 playing
cards. What is the probability that it is a queen or a heart?
Solution:
Q = Queen and H = Heart
P (Q ) =
4
13
1
, P (H ) = , P (Q I H ) =
52
52
52
P (Q U H ) = P (Q ) + P (H ) − P (Q I H )
=
12
4 13 1
16 4
+ −
=
=
52 52 52
52 13
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Conditional Probability
For any two events A and B with P(B) > 0, the conditional
probability
b bilit off A given
i
th
thatt B has
h occurred
d iis d
defined
fi d b
by
P (A | B ) =
P (A ∩ B )
P (B )
Which can be written:
P (A ∩ B ) = P (B ) ⋅ P (A | B )
Example:
Example: Consider the toss of two dice. Let
E = {sum of spots on dice is 4}
F = {sum of spots on dice is at most 4}.
P(E) = 1/12 since E = {(1, 3), (2, 2), (3, 1)}.
P(F) = 1/6 since
F = {(1, 1), (1, 2), (2, 1), (1, 3), (2, 2), (3, 1)}.
What about P(E | F)?
E = {sum of spots on dice is 4}
F = {sum of spots on dice is at most 4}
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P( E ∩ F )
P( F )
P( E )
=
P( F )
1
= 12
1
6
1
=
2
P( E F ) =
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Independence
Two events A and B are independent events if
Or
P (A | B ) = P (A ).
P(B / A) = P ( A)
Otherwise A and B are dependent.
Multiplicative Rule
If in an experiment the events A and B can both occur,
then
th
P( A ∩ B ) = P ( A) P ( B / A)
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Multiplicative Rule
Events A and B are independent events if and only if
P ( A ∩ B ) = P (A )P (B )
Note: this generalizes for more than two
independent events.
Example
One bag contains 4 white balls and 3 black balls, and a second bag
contains 3 white balls and 5 black balls. One ball is drawn from the
first bag and placed unseen in the second bag. What is the probability
that a ball now drawn from the second bag is black?
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Th k You
Thank
Y
Any Questions ?
STAT 319 – Probability and Statistics For Engineers
Dr Mohamed AICHOUNI
&
Dr Mustapha BOUKENDAKDJI
http://faculty.uoh.edu.sa/m.aichouni/stat319/
Email: [email protected]
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