© 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