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© 2002 Thomson / South-Western Slide 4B-1
Chapter 4,Part B
Probability
© 2002 Thomson / South-Western Slide 4B-2
Four Types of ProbabilityFour Types of Probability
• Marginal Probability
• Union Probability
• Joint Probability
• Conditional Probability
© 2002 Thomson / South-Western Slide 4B-3
Four Types of ProbabilityFour Types of Probability
Marginal
The probability of X occurring
Union
The probability of X or Y occurring
Joint
The probability of X and Y occurring
Conditional
The probability of X occurring given that Y has occurred
YX YX
Y
X
P X( ) P X Y( ) P X Y( ) P X Y( | )
© 2002 Thomson / South-Western Slide 4B-4
General Law of AdditionGeneral Law of Addition
P X Y P X P Y P X Y( ) ( ) ( ) ( )
YX
© 2002 Thomson / South-Western Slide 4B-5
General Law of Addition -- ExampleGeneral Law of Addition -- Example
P N S P N P S P N S( ) ( ) ( ) ( )
SN
.56 .67.70
P N
P S
P N S
P N S
( ) .
( ) .
( ) .
( ) . . .
.
70
67
56
70 67 56
0 81
© 2002 Thomson / South-Western Slide 4B-6
Office Design ProblemProbability Matrix
Office Design ProblemProbability Matrix
.11 .19 .30
.56 .14 .70
.67 .33 1.00
Increase Storage SpaceYes No Total
Yes
No
Total
Noise Reduction
© 2002 Thomson / South-Western Slide 4B-7
Office Design ProblemProbability Matrix, continued (2)
Office Design ProblemProbability Matrix, continued (2)
.11 .19 .30
.56 .14 .70
.67 .33 1.00
Increase Storage Space
Yes No TotalYes
No
Total
Noise Reduction
P N S P N P S P N S( ) ( ) ( ) ( )
. . .
.
70 67 56
81
© 2002 Thomson / South-Western Slide 4B-8
Office Design ProblemProbability Matrix, continued (3)
Office Design ProblemProbability Matrix, continued (3)
.11 .19 .30
.56 .14 .70
.67 .33 1.00
Increase Storage Space
Yes No TotalYes
No
Total
Noise Reduction
P N S( ) . . .
.
56 14 11
81
© 2002 Thomson / South-Western Slide 4B-9
Venn Diagram of the X or Y but Not Both Case
Venn Diagram of the X or Y but Not Both Case
YX
© 2002 Thomson / South-Western Slide 4B-10
Complement of a Union:The Neither/Nor RegionComplement of a Union:The Neither/Nor Region
YX
P X Y P X Y( ) ( ) 1
© 2002 Thomson / South-Western Slide 4B-11
Office Design Problem:The Neither/Nor RegionOffice Design Problem:The Neither/Nor Region
SN
P N S P N S( ) ( )
.
.
1
1 81
19
© 2002 Thomson / South-Western Slide 4B-12
Special Law of AdditionSpecial Law of Addition
If X and Y are mutually exclusive,
P X Y P X P Y( ) ( ) ( )
X
Y
© 2002 Thomson / South-Western Slide 4B-13
Demonstration Problem 4.3Demonstration Problem 4.3
Type of GenderPosition Male Female TotalManagerial 8 3 11Professional 31 13 44Technical 52 17 69Clerical 9 22 31Total 100 55 155
P T C P T P C( ) ( ) ( )
.
69
155
31
155645
© 2002 Thomson / South-Western Slide 4B-14
Demonstration Problem 4.3, continuedDemonstration Problem 4.3, continued
Type of GenderPosition Male Female TotalManagerial 8 3 11Professional 31 13 44Technical 52 17 69Clerical 9 22 31Total 100 55 155
P P C P P P C( ) ( ) ( )
.
44
155
31
155484
© 2002 Thomson / South-Western Slide 4B-15
Law of Multiplicationand Demonstration Problem 4.5
Law of Multiplicationand Demonstration Problem 4.5
P X Y P X P Y X P Y P X Y( ) ( ) ( | ) ( ) ( | )
P M
P S M
P M S P M P S M
( ) .
( | ) .
( ) ( ) ( | )
( . )( . ) .
80
1400 5714
0 20
0 5714 0 20 0 1143
© 2002 Thomson / South-Western Slide 4B-16
Special Law of Multiplication for Independent Events
Special Law of Multiplication for Independent Events
• General Law
• Special Law
P X Y P X P Y X P Y P X Y( ) ( ) ( | ) ( ) ( | )
If events X and Y are independent,
and P X P X Y P Y P Y X
Consequently
P X Y P X P Y
( ) ( | ), ( ) ( | ).
,
( ) ( ) ( )
© 2002 Thomson / South-Western Slide 4B-17
Law of Conditional ProbabilityLaw of Conditional Probability
• The conditional probability of X given Y is the joint probability of X and Y divided by the marginal probability of Y.
P X YP X Y
P Y
P Y X P X
P Y( | )
( )
( )
( | ) ( )
( )
© 2002 Thomson / South-Western Slide 4B-18
Law of Conditional Probability and the Office Design Problem
Law of Conditional Probability and the Office Design Problem
NS
.56 .70
P N
P N S
P S NP N S
P N
( ) .
( ) .
( | )( )
( )
.
..
70
56
56
7080
© 2002 Thomson / South-Western Slide 4B-19
Office Design Problem, continuedOffice Design Problem, continued
P N SP N S
P S( | )
( )
( )
.
..
11
67164
.19 .30
.14 .70
.33 1.00
Increase Storage SpaceYes No Total
YesNo
Total
Noise Reduction .11
.56
.67
© 2002 Thomson / South-Western Slide 4B-20
Independent EventsIndependent Events
• If X and Y are independent events, the occurrence of Y does not affect the probability of X occurring.
• If X and Y are independent events, the occurrence of X does not affect the probability of Y occurring.
If and are independent events,
( | ) ( ), and
( | ) ( ).
X Y
P X Y P X
P Y X P Y
© 2002 Thomson / South-Western Slide 4B-21
Revision of Probabilities: Bayes’ Rule
Revision of Probabilities: Bayes’ Rule
• Bayes’ Rule is an extension to the conditional law of probabilities
• Enables revision of original probabilities with new information
P X YP Y X P X
P Y X P X P Y X P X P Y X P Xi
i i
n n( | )
( | ) ( )
( | ) ( ) ( | ) ( ) ( | ) ( )
1 1 2 2
© 2002 Thomson / South-Western Slide 4B-22
Revision of Probabilities with Bayes' Rule: Ribbon Problem
Revision of Probabilities with Bayes' Rule: Ribbon Problem
P Alamo
P SouthJersey
P d Alamo
P d SouthJersey
P Alamo dP d Alamo P Alamo
P d Alamo P Alamo P d SouthJersey P SouthJersey
P SouthJersey dP d SouthJersey P SouthJersey
P d Alamo P Alamo P d SouthJersey P SouthJersey
( ) .
( ) .
( | ) .
( | ) .
( | )( | ) ( )
( | ) ( ) ( | ) ( )
( . )( . )
( . )( . ) ( . )( . ).
( | )( | ) ( )
( | ) ( ) ( | ) ( )
( . )( . )
( .
0 65
0 35
0 08
0 12
0 08 0 65
0 08 0 65 0 12 0 350 553
0 12 0 35
0 08)( . ) ( . )( . ).
0 65 0 12 0 350 447
© 2002 Thomson / South-Western Slide 4B-23
Revision of Probabilities with Bayes’ Rule: Ribbon Problem
Revision of Probabilities with Bayes’ Rule: Ribbon Problem
Conditional Probability
0.052
0.042
0.094
0.65
0.35
0.08
0.12
0.0520.094
=0.553
0.0420.094
=0.447
Alamo
South Jersey
Event
Prior Probability
P Ei( )
Joint Probability
P E di( )
Revised Probability
P E di( | )P d Ei( | )
© 2002 Thomson / South-Western Slide 4B-24
Revision of Probabilities with Bayes' Rule: Ribbon Problem
Revision of Probabilities with Bayes' Rule: Ribbon Problem
Alamo0.65
SouthJersey0.35
Defective0.08
Defective0.12
Acceptable0.92
Acceptable0.88
0.052
0.042
+ 0.094