Welcome to
EE315: Probabilistic Methods for
Electrical Engineers
Dr. Ali Hussein Muqaibel
Ver. 5.3
Dr. Ali Muqaibel
1
First Class Objectives:
• Welcome Students
• Who is your instructor? Office hours, contact information
• What is EE315?
•
•
•
•
Course resources
Blackboard
Syllabus: Content and Grading Policy
Calendar
Topics
Set Theory
Probability
• Why is this course is important?
• Way of thinking
• Communications: Bit and symbol decisions
• Power Systems: Demand and load forecasting
• Any question?
• Start with Set Theory
Random Variables (R. V.)
Operations on R. V.
Multiple R. V.
Operations on Multiple R. V.
Random Processes
Dr. Ali Muqaibel
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Probability & Random Processes for Electrical Engineers
Probability
Dr. Ali Hussein Muqaibel
Ver. 5.1
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Outlines
1. Set definitions
•
•
•
Element, Subset, Proper subset, Set, Class, Universal set=𝑆, empty=null set=𝜙.
Countable, uncountable, finite, infinite.
Disjoint, mutually exclusive.
2. Set operations
•
•
•
•
3.
4.
5.
6.
7.
8.
Venn diagram, equality, difference, union=sum, intersection=product, complement.
Algebra of sets (commutative, distributive, associative)
De Morgan’s law
Duality Principle
Probability introduced through sets and relative frequency
Joint and conditional probability
Independent events
Combined experiments
Bernoulli trials
Summary
Dr. Ali Muqaibel
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Probability & Random Processes for Engineers
Set definitions
Dr. Ali Hussein Muqaibel
Ver. 5.1
Dr. Ali Muqaibel
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Outlines (Set definitions)
Element, Set, Class
Set properties:
• Countable, uncountable
• Finite, infinite
• Subset & proper subset
• Universal set, empty=null set
• Two sets: Disjoint=mutually exclusive.
•
•
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Set Definition
Set
𝐴
𝑎
• Set: Collection of objects “elements”
• Class of sets: set of sets
• Notation:
elements
Class
• Set 𝐴, element 𝑎
• Notation: 𝑎 ∈ 𝐴 𝑜𝑟 𝑎 ∉ 𝐴
𝐴
𝐵
• How to define a set?
1.
2.
𝑎
Tabular {.}: Enumerate explicitly 6,7,8,9 .
Rule: {Integers between 5 and 10}.
𝐶
Set Definition Examples:
𝑍 = the set of integers
𝑅 = the set of real numbers
𝑁 = the set of natural numbers
𝑄 = the set of rational numbers
𝐴 = {2,4,6,8,10, ⋯ }
𝐴 = {𝑎 ∈ 𝑁| 𝑎 is even}
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Set Definitions: Countable/ finite
• Countable set: elements can be put in one-to-one correspondence with natural numbers 1,2,3,,…
Set of all real numbers between 0 and 1
• Not countable= uncountable.
0
1
• Set is finite: empty or counting its elements terminates i.e. finite number of elements
• Not finite: infinite.
• Finite => countable
• Uncountable => infinite
• Example: Describe
𝐴 = {2,4,6,8,10, ⋯ }
𝐴 = {𝑎 ∈ 𝑁| 𝑎 is even}
𝑍 = the set of integers
𝑁 = the set of natural numbers
𝑄 = the set of rational numbers
𝑅 = the set of real numbers
? Countable, uncountable, finite, infinite.
𝐴 = {2,4,6, . . . } is countable.
𝑁, 𝑍, & 𝑄 are countable.
𝑅 is uncountable.
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Universal Set & Null Set
• Universal set ( : all –encompassing set of objects under discussion.
• Example:
• Tossing two coins:
• Rolling a die:
• For any universal set with
subsets of
elements, there are
• Example: rolling a die, The universal set is
are
subset.
possible
, there
• Empty set=null set= has no elements
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Subset, Proper Subset, and Disjoint sets
𝑏
• The symbol denotes subset and denotes proper subset
•
Every element of is also an element in .
• Mathematically:
at least one element in
•
B
A
is not in .
• Statement: The null set is a subset of all other sets.
• Disjoint=Mutually exclusive: no common elements.
•
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A
B
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Set Definitions :Exercise
For the following sets ..specify:
(tabular/rule defined) (finite/infinite) (countable/uncountable)
Few examples:
is uncountable infinite
is tabular format
Next, we will do
some operations
𝐴∩𝐸 =𝜙
𝐷≠𝜙
Dr. Ali Muqaibel
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Probability & Random Processes for Engineers
Set Operations
Dr. Ali Hussein Muqaibel
Ver. 5.1
Objective: Venn Diagram, Equality & Difference, union and intersection,
complement, Algebra of sets, De Morgan’s laws, Duality principles.
Dr. Ali Muqaibel
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Outlines (Set operations)
•
•
•
•
•
Venn diagram,
Equality, difference, union=sum, intersection=product,
complement.
Algebra of sets (commutative, distributive, associative)
De Morgan’s law
Duality principle
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Set Operations
S
• Venn Diagram: sets are represented by closed-plane
figures. The universal sets is represented by a
rectangle.
• Equality:
“same elements”.
• Differences:
elements in that are not in .
• Example: Find
and
.
•
•
0.6
•
•
•
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B
A
C
1.6
1
2.5
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Union and Intersection
• Union=sum=
• Intersection=product=
A
B
C
𝐶∩𝐵
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Set Operations: Complement
• Complement
• Statements:
.
•
•
•
•
•
• Using the complement we can redefine the
difference as
.
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Set Operations: Algebra of sets
All subsets from an algebraic system
• Commutative
• Distributive
• Associative
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De Morgan’s Law
• De Morgan’s Law: the complement of a union (intersection) of two sets
and equals to the intersection (union) of the complements and .
• To complement an expression, we replace all sets by their complement, all unions
by intersections, and all intersections by union.
•
•
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Duality Principle
• Duality Principle: if in a set identity we replace all unions by
intersections, all intersections by unions, and the set and by
the sets and we get the dual identity.
•
•
• Example: Find the dual of the following identity
• By duality
(new dual identity)
• Exercise : Consider the two basic events
Dr. Ali Muqaibel
& .Simplify the following:
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Probability & Random Processes for Engineers
Probability Def. and Axioms
Dr. Ali Hussein Muqaibel
Ver. 5.1
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Probability Introduced through Sets and
Relative Frequency
• Objectives:
•
•
•
•
•
•
Experiments and sample spaces
Discrete and continuous sample spaces
Events
Probability definition and axioms
Mathematical model of experiments
Probability as a relative frequency
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Experiments and Sample Spaces
• For an experiment we define:
1. Sample Space 𝑆
2. Event Ω
3. Probability 𝑃
• Experiment: a single performance of an experiment is called a trial for which there is an outcome.
• Example: Rolling an unbiased (equally likely) die… likelihood 1/6 (probability)
• Sample Space 𝑆 = 1,2,3,4,5,6
• Prob. Set=
, , , , ,
• Discreet and Continuous Sample Spaces
• Die tossing => discrete and finite
• Positive integer=> discrete and infinite
• A pointer in a wheel 0 < 𝑠 ≤ 12 => continuous
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Events
• We are usually interested on the characteristics of the outcomes
rather than the outcomes themselves.
• Example: Draw a card from a deck of 52 cards.
• Events:
.
• Events defined on a countably infinite sample space do not have to
be countably infinite. Example
out of positive integers.
• Events defined on continuous spaces are usually continuous but can
be discrete.
spades (♠), hearts (♥), diamonds (♦) and clubs (♣)
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Probability Axioms and Def.
•
Def. of Probability
1. Set theory & fundamental axioms “sound”
2. Relative frequency “easy”
Probability has a nonnegative value. It is function of the event and it satisfies
three axioms.
• Axiom 1: 𝑃 𝐴 ≥ 0
• Axiom 2: 𝑃 𝑆 = 1, 𝑆 is the certain event.
• Axiom 3: 𝑃(⋃
𝐴 )=∑
𝑃 𝐴 𝑖𝑓 𝐴 ∩ 𝐴 = 𝜙 𝑓𝑜𝑟 𝑚 ≠ 𝑛 . The probability of the event
that equals to the union of any number of mutually exclusive event is equal to the sum of the
individual event probabilities.
• Based on common sense, engineering and scientific observation, we define
probability with “Relative frequency”
• Head and Tail of a coin 𝑃 𝐻 = lim
→
• Assuming “statistical regularity” like physical experiments.
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Continuous Outcomes
• Spinning the pointer on a “fair” wheel of chance
• What is the probability of the pointer falling between
• What if we divide the range into
?
contiguous segment?
• The probability of a discrete event defined on a continuous sample
space =0
• Event can occur even if their probability is 0 ?!
• (This is different than the impossible event)
• Event with probability 1 may not occur.
•
• This is different than the certain event.
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Example: Mathematical Model of Experiment
Mathematical Model of Experiment :Sample Space, Event, Probability.
Example: Sum of two dice, find the probability of 𝐴, 𝐵, 𝐶 , and 𝐵 ∪ 𝐶.
• 𝐴 = {𝑆𝑢𝑚 = 7}
• 𝐵 = {8 < 𝑠𝑢𝑚 ≤ 11}
• 𝐶 = {10 < 𝑠𝑢𝑚}
• There are 36 elementary events, 𝐴
• 𝑃(𝐴 ) = 1/36
• 𝑃 𝐴 = 𝑃(⋃
• 𝑃 𝐵 =
𝐴,
)=∑
𝑃 𝐴,
=6
=
=
• 𝑃 𝐶 =3
=
• 𝑃 𝐵∪𝐶 =
=
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Example: Probability Introduced Through Sets and Relative Frequency
Resistivity (Ohms)
10
22
27
47
#
18
12
33
17
• In a box, there are 80 resistors (same size and shape)
•
•
•
•
0
• Because they are mutually exclusive, a resistor must be chosen
• Suppose that the first is
(without replacement)
•
•
•
•
Condition is read as given that
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Probability & Random Processes for Engineers
Joint and Conditional
Probability
Dr. Ali Hussein Muqaibel
Ver. 5.1
Objective:
 Joint Probability
 Conditional Probability
 Total Probability
 Bayes Theorem
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Joint and Conditional Probability
•


 The probability of the union never exceeds the sum

Counted Twice
• Conditional Probability

( ∩ )
,
 If
 All the three axioms of probability holds true for conditional probability
 To show axiom 2: Let
∩
then
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Ω
Example: Joint and Conditional Probability
•
•
•
•
•
Draw a resistor from the box with the same likelihood
Event A “ a draw of
resister”
Event B “a draw of 5% resister”
Event C “a draw of a
resistor”
Find 𝑃 𝐴 , 𝑃 𝐵 , 𝑃 𝐶 , 𝑃 𝐴 ∩ 𝐵 , 𝑃 𝐴 ∩ 𝐶 , 𝑃 𝐵 ∩ 𝐶 ,
5%
10%
Total
22
10
14
24
47
28
16
44
100
24
8
32
Total
62
38
100
𝑃 𝐴 𝐵 , 𝑃 𝐴 𝐶 , 𝑃 (𝐵|𝐶)
• 𝑃 𝐴 =
•
•
•
, 𝑃 𝐵 =
, 𝑃 𝐶 =
∩
•
•
•
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Total Probability
Given 𝑁 mutually exclusive events
𝐵 , 𝑛 = 1,2, … . 𝑁, that divides the
universe
𝐵 ∩𝐵 =𝜙
for all 𝑚 ≠ 𝑛
𝐵 =𝑆
Total Probability (proof)
N
N
n 1
n 1
A  A  S  A  ( Bn )   ( A  Bn )
mutually exclusive
N
N
n 1
n 1
P ( A)  P[ ( A  Bn )]   P ( A  Bn )
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1.4 Joint and Conditional Probability
Bayes’ Theorem
𝑃 𝐴𝐵 𝑃 𝐵
𝑃 𝐵 𝐴 =
𝑃 𝐴
𝑃 𝐵 𝐴 =
𝑃 𝐵 ∩𝐴
𝑃 𝐴
𝑃 𝐴𝐵
𝑃 𝐴∩𝐵
𝑃 𝐵
𝑃 𝐵 𝐴 =
=
𝑃 𝐴𝐵 𝑃 𝐵
𝑃 𝐴
• Given one conditional probability,𝑃 𝐴 𝐵 , we can find the other, 𝑃(𝐵|𝐴)
• Another form using the total probability to represent 𝑃(𝐴)
𝑃 𝐵 𝐴 =
𝑃 𝐴𝐵 𝑃 𝐵
𝑃 𝐴|𝐵 )𝑃(𝐵 + ⋯ + 𝑃 𝐴|𝐵 )𝑃(𝐵
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Example: Total Prob. & Bayes’ Theorem
𝐵
𝐵
𝐴
𝐴
: 1 before the channel
: 0 before the channel
: 1 After the channel
: 0 After the channel
A priori probabilities
𝑃 𝐵 = 0.6,
𝑃 𝐵 = 0.4
Conditional Probabilities or transition prob.
𝑃 𝐴 𝐵 ,𝑃 𝐴 𝐵 ,𝑃 𝐴 𝐵 ,𝑃 𝐴 𝐵
Find 𝑃 𝐵 𝐴 𝑎 𝒑𝒐𝒔𝒕𝑟𝑜𝑖𝑜𝑟𝑖 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠
𝑃 𝐵 𝐴 ,𝑃 𝐵 𝐴 ,𝑃 𝐵 𝐴
Dr. Ali Muqaibel
Binary Symmetric Channel (BSC)
Ex 1.4-2:
33
Example: Solution
Find
𝑃(𝐵 𝐴 ) =
𝑃(𝐵 ∩ 𝐴 ) 𝑃(𝐴 𝐵 )𝑃(𝐵 ) 0.9 × 0.6 54
=
=
=
≃ 0.931
𝑃(𝐴 )
𝑃(𝐴 )
0.58
58
𝑃(𝐵 𝐴 ) =? = 1 − 𝑃(𝐵 𝐴 )
Look at examples in the text book 1.4.3 & 1.4.4
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Probability & Random Processes for Engineers
Independent Events
&
Combined Experiments
Dr. Ali Hussein Muqaibel
Ver. 5.1
Objective:
 Independent Events
 Combined Experiments
 Permutation and Combination
 Bernoulli Trials
 De Moivre-Laplace & Poisson Approximations
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Statistically independent : the probability of one
event is not affected by the occurrence of the other
event, 𝑃 𝐴 𝐵 = 𝑃 𝐴 , and 𝑃(𝐵|𝐴) = 𝑃(𝐵)
1.5 Independent Events
 Def: Two events
 &
&
are (statistically) independent if
independent
 What about
∩
with the complement ?
 𝐵 =𝐵∩𝑆 =𝐵∩ 𝐴∪𝐴 = 𝐵∩𝐴 ∪ 𝐵∩𝐴
 𝑃(𝐵) = 𝑃(𝐵 ∩ 𝐴) + 𝑃(𝐵 ∩ 𝐴)
 𝐴 & 𝐵 independent ⇒
 𝑃(𝐵 ∩ 𝐴) = 𝑃(𝐵) − 𝑃(𝐵 ∩ 𝐴) = 𝑃(𝐵) − 𝑃(𝐵)𝑃(𝐴)
= 𝑃(𝐵)[1 − 𝑃(𝐴)] = 𝑃(𝐵)𝑃(𝐴)
 𝐴 & 𝐵 independent
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Comments about independence
• Sufficient Condition for independence (not just necessary)
𝑃 𝐴∩𝐵 =𝑃 𝐴 𝑃 𝐵
• Independence simplify many things “Usually assumed based
on physics”.
• If 𝑃 𝐴 > 0 & 𝑃 𝐵 > 0, 𝑡ℎ𝑒𝑛
𝑃 𝐴 ∩ 𝐵 = 0 ⇒ 𝑀𝑢𝑡𝑢𝑎𝑙𝑙𝑦 𝑒𝑥𝑐𝑙𝑢𝑠𝑖𝑣𝑒
𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 𝑃(𝐵) independent
• For two events to be independent 𝐴 ∩ 𝐵 ≠ 𝜙, because if they
are disjoint occurrence of one exclude the other.
• Example:
•
•
•
•
𝑨 “select a king”
𝑩 “select a jack or queen”
𝑪 “select a heart”
Find 𝑷 𝑨 , 𝑷 𝑩 , 𝑷 𝑪 , 𝑷 𝑨 ∩ 𝑩 , 𝑷 𝑨 ∩ 𝑪 , 𝑷 𝑩 ∩ 𝑪 and test
dependence.
Dr. Ali Muqaibel
4
52
8
𝑃(𝐵) =
52
13
𝑃(𝐶) =
52
𝑃 𝐴 =
32
𝑃 𝐴∩𝐵 =0≠𝑃 𝐴 𝑃 𝐵 =
52
1
𝑃 𝐴∩𝐶 =
=𝑃 𝐴 𝑃 𝐶
52
2
𝑃 𝐵∩𝐶 =
=𝑃 𝐵 𝑃 𝐶
52
𝐴 is independent of 𝐶 and
𝐵 is independent of 𝐶 , but
𝐴 & 𝐵 are dependent.
37
1.5 Independent Events
Multiple Events
• Def: 3 events
Independence by pairs is not sufficient all
combinations must be satisfied
are independent
• Example: Chosing a number in
• 𝑃 𝐴 ∩ 𝐴 = 𝑃 𝐴 𝑃 𝐴 , 𝑖 ≠ 𝑗, ⇒ 𝐴 , 𝐴 , and 𝐴 pairwise independent
• 𝑃 𝐴 ∩ 𝐴 ∩ 𝐴 ≠ 𝑃 𝐴 𝑃 𝐴 𝑃 𝐴 , ⇒ 𝐴 , 𝐴 , and 𝐴 NOT independent
Fact: A1 , A2 , & A3 independent  A1 & (A2  A3 ) independent
Fact: A1 , A2 , & A3 independent  A1 & (A2  A3 ) independent
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Comments about independence of
multiple events
• If 𝑁 events are independent, then any one of them is independent of any event formed by unions,
intersection, and complements of the others.
𝐴 is independent 𝐴 => 𝐴 is independent 𝐴
• If independent, intersection is changed to product
𝑃 𝐴 ∩ 𝐴 ∩𝐴
=𝑃 𝐴 𝑃 𝐴 ∩𝐴 =𝑃 𝐴 𝑃 𝐴 𝑃 𝐴
𝑃 𝐴 ∩ 𝐴 ∪𝐴 =𝑃 𝐴 𝑃 𝐴 ∪𝐴
• This is assuming full independence. Pairwise is not sufficient.
Example: Drawing four ace with replacement and without replacement
With replacement 𝑃 𝐴 ∩ 𝐴 ∩ 𝐴 ∩ 𝐴
= 𝑃(𝐴 )𝑃(𝐴 ) 𝑃(𝐴 ) 𝑃(𝐴 ) =
Without replacement 𝑃 𝐴 ∩ 𝐴 ∩ 𝐴 ∩ 𝐴
𝐴 ) = × × × ≈ 3.69(10 )
≈ 3.5 10
= 𝑃(𝐴 )𝑃(𝐴 |𝐴 ) 𝑃(𝐴 |𝐴 ∩ 𝐴 ) 𝑃(𝐴 |𝐴 ∩ 𝐴 ∩
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Combined Experiments
• Combined experiments are made of sub-experiments.
• Examples (Wind speed, pressure)
𝑦
𝐴×𝑆
𝑦
𝑆
𝐵
𝐴×𝐵
𝑆 ×𝐵
𝑦
𝑥
• Repeating an exp. 𝑁 times: Flipping a coin 𝑁 times.
𝑆
𝐴
𝑥
𝑥
𝑆
• For two independent (𝑆 , Ω , 𝑃 ) with 𝑁 elements and (𝑆 , Ω , 𝑃 ) with 𝑀 elements, we can form a combined
experiment (𝑆, Ω, 𝑃) with 𝑁𝑀 elements where 𝑆 = 𝑆 × 𝑆 = 𝑠 , 𝑠 , 𝑠 ∈ 𝑆 , 𝑠 ∈ 𝑆
• Example 1: 𝑆 = 𝐻, 𝑇 , 𝑆 = {1,2,3,4,5,6}
• 𝑆 = { 𝑇, 1 , 𝑇, 2 , 𝑇, 3 , 𝑇, 4 , 𝑇, 5 , 𝑇, 6 , 𝐻, 1 , 𝐻, 2 , 𝐻, 3 , 𝐻, 4 , 𝐻, 5 , 𝐻, 6 }
• Example 2: 𝑆 = 𝐻, 𝑇 , 𝑆 = 𝐻, 𝑇
• 𝑆=
𝐻, 𝐻 , 𝐻, 𝑇 , 𝑇, 𝐻 , 𝑇, 𝑇
• Events on the combined Space, 𝑠 ∈ 𝐴, 𝑠 ∈ 𝐵, 𝐶 = 𝐴 × 𝐵
• 𝐴×𝐵 = 𝐴×𝑆
∩ (𝑆 × 𝐵)
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Probability of Combined Experiments
• 3 independent experiments 𝑆 , Ω , 𝑃 , 𝑖 = 1,2,3 can define a combined probability space 𝑆, Ω, 𝑃 :
• 𝑆 =𝑆 ×𝑆 ×𝑆
• Ω =Ω ×Ω ×Ω
• 𝑃 𝐴 ×𝐴 ×𝐴
=𝑃 𝐴 𝑃 𝐴 𝑃 𝐴 ,
𝐴 ∈Ω
• Permutation: # of possible sequences (order important) (not replaced)
𝑛!
𝑃 =𝑛 𝑛−1 … 𝑛−𝑟+1 =
𝑛−𝑟 !
• Combination: # of possible sequences (order not important) (not replaced), # decreases by 𝑃 =
𝑛
=𝐶 =
𝑟
!
!
= 𝑟!
!
!
!
• A Permutation is an ordered Combination
𝑛
• The binomial expansion 𝑥 + 𝑦 = ∑
𝑥 𝑦
𝑟
• Example for 𝑆 = {A, B, C, D, E}
Dr. Ali Muqaibel
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Examples: Permutation and Combination
• Example: drawing 4 cards from 52 card deck. How many
permutations are there.
• Example: A coach has five athletes and he wants to make a team
made of 3. How many teams can he make?
!
! !
Same as choosing 2 for the spare team.
Other notations are also possible.
Dr. Ali Muqaibel
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Bernoulli Trials
Hit or miss , win or lose, 0 or 1
Basic experiment with 2 possible outcomes
and
Benoulli trials repeat the basic experiment times
(Assume that elementary events are independent for every trial.)
Example: We are firing a carrier with torpedoes.
two or more hits. We are firing three torpedoes.
,
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. It will sunk if
43
Continue Example :Bernoulli Trials
3
P (0 hits)    0.40 (1  0.4)3  0.216
 0
 3
P (1 hits)    0.41 (1  0.4) 2  0.432
1 
 3
P (3 hits)    0.43 (1  0.4)0  0.064
 3
P ({carrier sunk})  P (2 hits)  P (3 hits)  0.352
Example: Given we are firing for 3 seconds. Firing rate 2400 per minutes. Find 𝑃{𝑒𝑥𝑎𝑐𝑡𝑙𝑦 50 ℎ𝑖𝑡𝑠}
N  120
120  50
70
P (50 hits)  
0.4
(1

0.4)
?

50


P( A)  0.4
large N 
𝑁 = 3𝑠𝑒𝑐 ×
2400𝑏𝑢𝑙𝑙𝑒𝑡𝑠/𝑚𝑖𝑛
= 120 𝑏𝑢𝑙𝑙𝑒𝑡𝑠.
60 𝑠𝑒𝑐/𝑚𝑖𝑛
120!  ?
De Moivre-Laplace approximation
Poisson approximation
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44
De Moivre-Laplace & Poisson Approximations
• Stirling’s Formula:
for large
• Error less than 1% even for 𝑚 = 10.
• Using Stirling’s formula, De Moivre-Laplace Approximation
• 𝑁, 𝑘, 𝑎𝑛𝑑 𝑁 − 𝑘, 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑙𝑎𝑟𝑔𝑒 , 𝑘 𝑚𝑢𝑠𝑡 𝑏𝑒 𝑛𝑒𝑎𝑟 𝑁𝑝 to assure small numerator.
• If is very large and is very small De Moivre-Laplace approximation fails, we can use
Poisson approximation:
!
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large and
is small.
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Example: De Moivre-Laplace Approximation
Back to the torpedoes example. Given we are firing for 3 seconds. Firing rate 2400
per minutes. Find
/
•
/
•
are all large.
•
•
is near
=
Dr. Ali Muqaibel
46
In Class Practice: Review of Combined Experiments and Repeated Trials
One of the First Problems Solved by Pascal
• A pair of dice is rolled
times.
 Find the probability that “seven” will not show up at all.
 The probability of obtaining double-six at least once.
 Find the number of throws required to assure more than 50% success of
obtaining double-six at least once.
Dr. Ali Muqaibel EE570
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Solution to the in class practice
• A pair of dice is rolled 𝑛 times.
 Find the probability that “seven” will not show up at all.
 𝐴 = 𝑠𝑒𝑣𝑒𝑛 = 1,6 , 2,5 , 3,4 , 4,3 , 5,2 , 6,1
 𝑃 𝐴 =
= , 𝑃 𝐴̅ = , 𝑃 0 =
 The probability of obtaining double-six at least once.
 𝐵 = 𝑑𝑜𝑢𝑏𝑙𝑒 𝑠𝑖𝑥 = 6,6
 𝑃 𝐵 =
,𝑃 𝐵 =
 𝑋 = {𝑑𝑜𝑢𝑏𝑙𝑒 𝑠ix at least once in 𝑛 times}
 𝑋 ={double six will not show up in any of the trials}= 𝐵𝐵 𝐵…. 𝐵
 𝑃 𝑋 = 1−𝑃 𝑋 = 1−
 Find the number of throws required to assure more than 50% success of obtaining double-six at least once.
 1−
>
 𝑛 log
< − log 2 or 𝑛 > log 2/(log 36 − log 35) = 24.605
𝑜𝑟
<
 The answer 𝑛 = 25 throws
Dr. Ali Muqaibel EE570
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