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Name Shakeel Nouman Religion Christian Domicile Punjab (Lahore) Contact # 0332-4462527. 0321-9898767 E.Mail [email protected] [email protected]
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Slide 1
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Shakeel NoumanM.Phil Statistics
Probability
Slide 2
Using Statistics Basic Definitions: Events, Sample Space,
and Probabilities Basic Rules for Probability Conditional Probability Independence of Events Combinatorial Concepts The Law of Total Probability and Bayes’
Theorem Summary and Review of Terms
Probability2
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 32-1 Probability is:
A quantitative measure of uncertainty A measure of the strength of belief in the
occurrence of an uncertain event A measure of the degree of chance or likelihood
of occurrence of an uncertain event Measured by a number between 0 and 1 (or
between 0% and 100%)
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 4Types of Probability
Objective or Classical Probabilitybased on equally-likely eventsbased on long-run relative frequency of eventsnot based on personal beliefs is the same for all observers (objective) examples: toss a coin, throw a die, pick a card
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 5Types of Probability (Continued)
Subjective Probabilitybased on personal beliefs, experiences, prejudices, intuition -
personal judgmentdifferent for all observers (subjective) examples: Super Bowl, elections, new product introduction,
snowfall
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 62-2 Basic Definitions
Set - a collection of elements or objects of interestEmpty set (denoted by )
a set containing no elementsUniversal set (denoted by S)
a set containing all possible elementsComplement (Not). The complement of A is
a set containing all elements of S not in A
A
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 7Complement of a Set
A
A
S
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 8Basic Definitions (Continued)
Intersection (And)– a set containing all elements in both
A and BUnion (Or)
– a set containing all elements in A or B or both
A B
A B
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 9
A B
Sets: A Intersecting with B
AB
S
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 10Sets: A Union B
A B
AB
S
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 11Basic Definitions (Continued)
•Mutually exclusive or disjoint sets–sets having no elements in common,
having no intersection, whose intersection is the empty set
•Partition–a collection of mutually exclusive sets
which together include all possible elements, whose union is the universal set
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 12Mutually Exclusive or Disjoint
Sets
A B
S
Sets have nothing in common
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 13Sets: Partition
1A
2A
3A
4A
5A
S
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 14Experiment
• Process that leads to one of several possible outcomes *, e.g.:Coin toss
» Heads,TailsThrow die
» 1, 2, 3, 4, 5, 6Pick a card
» AH, KH, QH, ...Introduce a new product
• Each trial of an experiment has a single observed outcome.
• The precise outcome of a random experiment is unknown before a trial.
* , , Also called a basic outcome elementary event or simple event
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 15Events : Definition
Sample Space or Event SetSet of all possible outcomes (universal set) for a
given experiment E.g.: Throw die
• S = {1,2,3,4,5,6} Event
Collection of outcomes having a common characteristic
E.g.: Even number • A = {2,4,6}
– Event A occurs if an outcome in the set A occurs Probability of an event
Sum of the probabilities of the outcomes of which it consists
P(A) = P(2) + P(4) + P(6)
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 16Equally-likely Probabilities
(Hypothetical or Ideal Experiments)
• For example:Throw a die
» Six possible outcomes {1,2,3,4,5,6}» If each is equally-likely, the probability of each is 1/6
= .1667 = 16.67%
» » Probability of each equally-likely outcome is 1 over
the number of possible outcomesEvent A (even number)
» P(A) = P(2) + P(4) + P(6) = 1/6 + 1/6 + 1/6 = 1/2» for e in AP A P e
n A
n S
( ) ( )
( )
( )
3
6
1
2
P en S
( )( )
1
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 17
Hearts Diamonds Clubs Spades
A A A AK K K KQ Q Q QJ J J J
10 10 10 109 9 9 98 8 8 87 7 7 76 6 6 65 5 5 54 4 4 43 3 3 32 2 2 2
‘ ’Event Ace Union of ‘ ’Events Heart ‘ ’and Ace
‘ ’Event Heart
The intersection of the ‘ ’ ‘ ’ events Heart and Ace
comprises the single point : circled twice the ace of hearts
P Heart Ace
n Heart Ace
n S
( )
( )
( )
16
52
4
13
P Heartn Heart
n S
( )( )
( )
13
52
1
4
P Acen Ace
n S
( )( )
( )
4
52
1
13
P Heart Acen Heart Ace
n S
( )( )
( )
1
52
Pick a Card: Sample Space
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 182-3 Basic Rules for Probability
Range of Values
Complements - Probability of not A
Intersection - Probability of both A and B
Mutually exclusive events (A and C) :
Range of Values
Complements - Probability of not A
Intersection - Probability of both A and B
Mutually exclusive events (A and C) :
0 1 P A( )
P A P A( ) ( ) 1
P A B n A Bn S
( ) ( )( )
P A C( ) 0
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 19 Basic Rules for Probability
(Continued)
• Union - Probability of A or B or both (rule of unions)
Mutually exclusive events: If A and B are mutually exclusive, then
• Union - Probability of A or B or both (rule of unions)
Mutually exclusive events: If A and B are mutually exclusive, then
P A B n A Bn S
P A P B P A B( ) ( )( )
( ) ( ) ( )
)()()( 0)( BPAPBAPsoBAP
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 20Sets: P(A Union B)
)( BAP
AB
S
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 21 Basic Rules for Probability
(Continued)
• Conditional Probability - Probability of A given B
Independent events:
• Conditional Probability - Probability of A given B
Independent events:
0)( ,)(
)()( BPwhereBP
BAPBAP
P A B P A
P B A P B
( ) ( )
( ) ( )
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 22
:Rules of conditional probability :Rules of conditional probability
If events A and D are statistically independent:
so
so
P A B P A BP B
( ) ( )( )
P A B P A B P B
P B A P A
( ) ( ) ( )
( ) ( )
P AD P A
P D A P D
( ) ( )
( ) ( )
P A D P A P D( ) ( ) ( )
2-4 Conditional Probability
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 23
AT& T IBM Total
Telecommunication 40 10 50
Computers 20 30 50
Total 60 40 100
Counts
AT& T IBM Total
Telecommunication .40 .10 .50
Computers .20 .30 .50
Total .60 .40 1.00
Probabilities
P IBM TP IBM T
P T( )
( )
( )
.
..
10
502
Probability that a project is undertaken by IBM
given it is a telecommunications
project:
Contingency Table - Example 2-2
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 24
P A B P A
P B A P B
and
P A B P A P B
( ) ( )
( ) ( )
( ) ( ) ( )
Conditions for the statistical independence of events A and B:
P Ace HeartP Ace Heart
P Heart
P Ace
( )( )
( )
( )
1521352
113
P Heart AceP Heart Ace
P Ace
P Heart
( )( )
( )
( )
1524
52
14
P Ace Heart P Ace P Heart( ) ( ) ( ) 4
521352
152
2-5 Independence of Events
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 25
a P T B P T P B
b P T B P T P B P T B
) ( ) ( ) ( )
. * . .
) ( ) ( ) ( ) ( )
. . . .
0 04 0 06 0 0024
0 04 0 06 0 0024 0 0976
Events Television (T) and Billboard (B) are assumed to be independent.
Independence of Events - Example 2-5
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 26
The probability of the union of several independent events is 1 minus the product of probabilities of their complements:
P A A A An P A P A P A P An( ) ( ) ( ) ( ) ( )1 2 3
11 2 3
Example 2-7:( ) ( ) ( ) ( ) ( )
. . .
Q Q Q Q P Q P Q P Q P Q1 2 3 10
11 2 3 10
1 9010 1 3487 6513
The probability of the intersection of several independent events is the product of their separate individual probabilities:
P A A A An P A P A P A P An( ) ( ) ( ) ( ) ( )1 2 3 1 2 3
Product Rules for Independent Events
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 27
Consider a pair of six-sided dice. There are six possible outcomes from throwing the first die {1,2,3,4,5,6} and six possible outcomes from throwing the second die {1,2,3,4,5,6}. Altogether, there are
6*6=36 possible outcomes from throwing the two dice.
In general, if there are n events and the event i can happen in Ni possible ways, then the number of ways in which the
sequence of n events may occur is N1N2...Nn.
2-6 Combinatorial Concepts
Pick 5 cards from a deck of 52 - with replacement
52*52*52*52*52=525 380,204,032 different possible outcomes
Pick 5 cards from a deck of 52 - without replacement
52*51*50*49*48 = 311,875,200 different possible outcomes
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 28
.
..
. .Order the letters: A, B, and C
A
B
C
B
C
A
B
A
C A
C
B
C
B
A
. ....
.
..
..
. ABC
ACB
BAC
BCA
CAB
CBA
More on Combinatorial Concepts(Tree Diagram)
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 29
How many ways can you order the 3 letters A, B, and C?
There are 3 choices for the first letter, 2 for the second, and 1 for the last, so there are 3*2*1 = 6 possible ways to order the three
letters A, B, and C.
How many ways are there to order the 6 letters A, B, C, D, E, and F? (6*5*4*3*2*1 = 720)
Factorial: For any positive integer n, we define n factorial as:n(n-1)(n-2)...(1). We denote n factorial as n!.
The number n! is the number of ways in which n objects can be ordered. By definition 1! = 1 and 0! = 1.
Factorial
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 30
Permutations are the possible ordered selections of r objects out of a total of n objects. The number of permutations of n objects
taken r at a time is denoted by nPr, where
What if we chose only 3 out of the 6 letters A, B, C, D, E, and F?There are 6 ways to choose the first letter, 5 ways to choose the
second letter, and 4 ways to choose the third letter (leaving 3letters unchosen). That makes 6*5*4=120 possible orderings or
permutations.
n Prn
n r
For example
P
!( )!
:
!
( )!
!
!
* * * * *
* ** *
6 3
6
6 3
6
3
6 5 4 3 2 1
3 2 16 5 4 120
Permutations (Order is important)
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 31
Combinations are the possible selections of r items from a group of n itemsregardless of the order of selection. The number of combinations is denoted
and is read as n choose r. An alternative notation is nCr. We define the numberof combinations of r out of n elements as:
Suppose that when we pick 3 letters out of the 6 letters A, B, C, D, E, and F we chose BCD, or BDC, or CBD, or CDB, or DBC, or DCB. (These are the6 (3!) permutations or orderings of the 3 letters B, C, and D.) But these are
orderings of the same combination of 3 letters. How many combinations of 6different letters, taking 3 at a time, are there?
n
rC
n!
r!(n r)!
n
r
n r
For example
C
:
!
!( )!
!
! !
* * * * *
( * * )( * * )
* *
* *6 3
6
3 6 3
6
3 3
6 5 4 3 2 1
3 2 1 3 2 1
6 5 4
3 2 1
120
620
n
r
Combinations (Order is not Important)
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 32Example: Template for Calculating
Permutations & Combinations
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 33
P A P A B P A B( ) ( ) ( )
In terms of conditional probabilities:
More generally (where Bi make up a partition):
P A P A B P A BP A B P B P A B P B
( ) ( ) ( )( ) ( ) ( ) ( )
P A P A Bi
P ABi
P Bi
( ) ( )
( ) ( )
2-7 The Law of Total Probability and Bayes’ Theorem
The law of total probability:
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 34
Event U: Stock market will go up in the next yearEvent W: Economy will do well in the next year
P UW
P U W
P W P W
P U P U W P U WP U W P W P U W P W
( ) .
( )
( ) . ( ) . .
( ) ( ) ( )( ) ( ) ( ) ( )
(. )(. ) (. )(. ). . .
75
30
80 1 8 2
75 80 30 2060 06 66
The Law of Total Probability-Example 2-9
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 35Bayes’ Theorem
• Bayes’ theorem enables you, knowing just a little more than the probability of A given B, to find the probability of B given A.
• Based on the definition of conditional probability and the law of total probability.
P B AP A B
P A
P A B
P A B P A B
P AB P B
P AB P B P AB P B
( )( )
( )
( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
Applying the law of total probability to the denominator
Applying the definition of conditional probability throughout
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 36Bayes’ Theorem - Example 2-10
• A medical test for a rare disease (affecting 0.1% of the population [ ]) is imperfect:When administered to an ill person, the test will indicate
so with probability 0.92 [ ]» The event is a false negative
When administered to a person who is not ill, the test will erroneously give a positive result (false positive) with probability 0.04 [ ] » The event is a false positive. .
P I( ) .0 001
08.)(92.)( IZPIZP
)( IZ
)( IZ
96.0)(04.0)( IZPIZP
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 37
P I
P I
P Z I
P Z I
( ) .
( ) .
( ) .
( ) .
0001
0999
092
004
P I ZP I Z
P Z
P I Z
P I Z P I Z
P Z I P I
P Z I P I P Z I P I
( )( )
( )
( )
( ) ( )
( ) ( )
( ) ( ) ( ) ( )
(. )( . )
(. )( . ) ( . )(. )
.
. .
.
..
92 0001
92 0001 004 999
000092
000092 003996
000092
040880225
Example 2-10 (continued)
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 38
P I( ) .0001
P I( ) .0999 P Z I( ) .004
P Z I( ) .096
P Z I( ) .008
P Z I( ) .092 P Z I( ) ( . )( . ) . 0 001 0 92 00092
P Z I( ) ( . )( . ) . 0 001 0 08 00008
P Z I( ) ( . )( . ) . 0 999 0 04 03996
P Z I( ) ( . )( . ) . 0 999 0 96 95904
Prior Probabilities
Conditional Probabilities
JointProbabilities
Example 2-10 (Tree Diagram)
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 39
• Given a partition of events B1,B2 ,...,Bn:
P B AP A B
P A
P A B
P A B
P A B P B
P A B P B
i
i i
( )( )
( )
( )
( )
( ) ( )
( ) ( )
1
1
1
1 1
Applying the law of total probability to the denominator
Applying the definition of conditional probability throughout
Bayes’ Theorem Extended
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 40
An economist believes that during periods of high economic growth, the U.S. dollar appreciates with probability 0.70; in periods of moderate economic growth, the dollar appreciates with probability 0.40; and during periods of
low economic growth, the dollar appreciates with probability 0.20. During any period of time, the probability of high economic growth is 0.30,
the probability of moderate economic growth is 0.50, and the probability of low economic growth is 0.50.
Suppose the dollar has been appreciating during the present period. What is the probability we are experiencing a period of high economic growth?
Partition:H - High growth P(H) = 0.30
M - Moderate growth P(M) = 0.50L - Low growth P(L) = 0.20
Event A Appreciation
P A HP A MP A L
( ) .( ) .( ) .
0 700 40
0 20
Bayes’ Theorem Extended -Example 2-11
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 41
P H AP H A
P AP H A
P H A P M A P L AP A H P H
P A H P H P A M P M P A L P L
( )( )
( )( )
( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( . )( . )
( . )( . ) ( . )( . ) ( . )( . ).
. . ...
.
0 70 0 300 70 0 30 0 40 050 0 20 0 20
0 210 21 0 20 0 04
0 210 45
0 467
Example 2-11 (continued)
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 42
Prior Probabilities
Conditional Probabilities
JointProbabilities
P H( ) . 0 30
P M( ) . 0 50
P L( ) . 0 20
P A H( ) . 0 70
P A H( ) . 0 30
P A M( ) .0 40
P A M( ) . 0 60
P A L( ) . 0 20
P A L( ) . 0 80
P A H( ) ( . )( . ) . 0 30 0 70 0 21
P A H( ) ( . )( . ) . 0 30 0 30 0 09
P A M( ) ( . )( . ) . 0 50 0 40 0 20
P A M( ) ( . )( . ) . 0 50 0 60 0 30
P A L( ) ( . )( . ) . 0 20 0 20 0 04
P A L( ) ( . )( . ) . 0 20 0 80 0 16
Example 2-11 (Tree Diagram)
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 43
2-8 Using Computer: Template for Calculating the Probability
of at least one success
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer
Slide 44
M.Phil (Statistics)
GC University, . (Degree awarded by GC University)
M.Sc (Statistics) GC University, . (Degree awarded by GC University)
Statitical Officer(BS-17)(Economics & Marketing Division)
Livestock Production Research Institute Bahadurnagar (Okara), Livestock & Dairy Development
Department, Govt. of Punjab
Name Shakeel NoumanReligion ChristianDomicile Punjab (Lahore)Contact # 0332-4462527. 0321-9898767E.Mail [email protected] [email protected]
Probability By Shakeel Nouman M.Phil Statistics Govt. College University Lahore, Statistical Officer