Ch04.Probability

Business Statistics, 4eby Ken Black

Chapter 4Probability

A LOTTERY IS A TAX ON PEOPLE

WHO DON'T UNDERSTAND STATISTICS!!

Discrete Distributions

Learning Objectives

• Comprehend the different ways of assigning probability.

• Understand and apply marginal, union, joint, and conditional probabilities.

• Select the appropriate law of probability to use in solving problems.

• Solve problems using the laws of probability including the laws of addition, multiplication and conditional probability

• Revise probabilities using Bayes’ rule.

Methods of Assigning Probabilities

• Applications: chances of having break downs, defective items, customer visits in an hour of a day, life insurance claims, accidents, employees attrition, life of a mechanical part; company’s hiring practices meeting govt. criteria; managing risk; regulators examining whether criteria is met.

• Classical method of assigning probability (rules and laws)

• Relative frequency of occurrence (cumulated historical data)

• Subjective Probability (personal intuition or reasoning)

Classical Probability• Method of assigning

probability based on laws: involves a random experiment, outcomes, events. Probability can be determined even before the experiment.

• Number of outcomes leading to the event divided by the total number of outcomes possible

• Each outcome is equally likely• Applicable to games of chance• Objective -- everyone correctly

using the method assigns an identical probability

total number of outcomes

number of outcomes in Een

Relative Frequency Probability

• Based on historical data

• Computed after performing the experiment

• Number of times an event occurred divided by the number of trials

• Objective -- everyone correctly using the method assigns an identical probability

total number of trials

number of outcomes

producing Een

Subjective Probability• Comes from a person’s intuition or reasoning• Subjective -- different individuals may assign

different numeric probabilities to the same event (farmer, oil price, success of a rural marketing campaign)

• Degree of belief (probability that a CEO will be kidnapped by an unhappy employee/ terrorist, chances of strike by trade, manager of a casino business will not pay tax, )

• Useful for unique (single-trial) experiments– New product introduction– Initial public offering of common stock– Site selection decisions– Sporting events

Structure of Probability• Experiment: random check on quality of a

product, interviewing 30 customers for their satisfaction with a gadget (tossing coin 2x, a die)

• Event: an outcome or collective outcomes (even#)• Elementary Events: event impossible to break

down into other events • Sample Space: list of all possible outcomes• Unions and Intersections• Mutually Exclusive Events• Independent Events• Collectively Exhaustive Events• Complementary Events

Experiment• Experiment: a process that produces outcomes

– More than one possible outcome– Exs: interview randomly selected 20 managers on

acquiring tech, selecting 50 radars for their efficiency; testing side effect of a drug

• Trial: one repetition of the process• Elementary Event: cannot be decomposed or

broken down into other events (outcome)• Event: an outcome of an experiment

– may be an elementary event, or– may be an aggregate of elementary events– usually represented by an uppercase letter, e.g., A,

An Example Experiment

Experiment: randomly select, without replacement, two families from the residents of Tiny Town

Elementary Event: the sample includes families A and C

Event: each family in the sample has children in the household

Event: the sample families own a total of four automobiles

Family Children in Household

Number of Automobiles

YesYesNoYes

Tiny Town Population

Sample Space

• The set of all elementary events for an experiment

• Methods for describing a sample space– roster or listing– tree diagram– set builder notation– Venn diagram

Sample Space: Roster Example• Experiment: randomly select, without

replacement, two families from the residents of Tiny Town

• Each ordered pair in the sample space is an elementary event, for example -- (D,C)

Family Children in Household

Number of Automobiles

YesYesNo

Listing of Sample Space

(A,B), (A,C), (A,D),(B,A), (B,C), (B,D),(C,A), (C,B), (C,D),(D,A), (D,B), (D,C)

Sample Space: Tree Diagram for Random Sample of Two Families

Sample Space: Set Notation for Random Sample of Two Families

• S = {(x,y) | x is the family selected on the first draw, and y is the family selected on the second draw}

• Concise description of large sample spaces

Sample Space

• Useful for discussion of general principles and concepts

Listing of Sample Space

(A,B), (A,C), (A,D),(B,A), (B,C), (B,D),(C,A), (C,B), (C,D),(D,A), (D,B), (D,C)

Venn Diagram

Union of Sets• The union of two sets contains an instance

of each element of the two sets.

1 4 7 9

2 3 4 5 6

1 2 3 4 5 6 7 9

, , , ,

, , , , , , ,

C IBM DEC Apple

F Apple Grape Lime

C F IBM DEC Apple Grape Lime

, , , ,

Intersection of Sets• The intersection of two sets contains only

those element common to the two sets.

1 4 7 9

2 3 4 5 6

, , , ,

C IBM DEC Apple

F Apple Grape Lime

C F Apple

Mutually Exclusive Events

• Events with no common outcomes

• Occurrence of one event precludes the occurrence of the other event

2 3 4 5 6

, , , ,

C IBM DEC Apple

F Grape Lime

P X Y( ) 0

Independent Events

• Occurrence of one event does not affect the occurrence or nonoccurrence of the other event

• The conditional probability of X given Y is equal to the marginal probability of X.

• The conditional probability of Y given X is equal to the marginal probability of Y.

)()|( XPYXP Prob of X when Y is known is same as when Y is unknownMeans Y doesn’t affect probability of X.

Collectively Exhaustive Events

• Contains all elementary events for an experiment

E1 E2 E3

Sample Space with three collectively exhaustive events

Complementary Events

• All elementary events not in the event ‘A’ are in its complementary event.

SampleSpace A

P Sample Space( ) 1

P A P A( ) ( ) 1A

Four Types of Probability

• Marginal Probability• Union Probability• Joint Probability• Conditional Probability

Four Types of Probability

Marginal

The probability of X occurring

The probability of X or Y occurring

The probability of X and Y occurring

Conditional

The probability of X occurring given that Y has occurred

P X( ) P X Y( ) P X Y( ) P X Y( | )

General Law of Additionfour laws of probability to solve prob problems: addition, conditional, multiplication and Bayes’ rule; no one rule but some rules easier than

others

P X Y P X P Y P X Y( ) ( ) ( ) ( )

Example

• In a survey of interior design workers were asked which changes in office design would increase productivity. No 1 change voted by 70% involved reducing noise, 67% said more storage and filing space. If one worker is selected randomly what is the prob that this person would select reducing noise or more filing space if 56% said that both would improve productivity?

General Law of Addition -- Example

P N S P N P S P N S( ) ( ) ( ) ( )

.56 .67.70

( ) . . .

70 67 56

Office Design ProblemProbability Matrix

.11 .19 .30

.56 .14 .70

.67 .33 1.00

Increase Storage SpaceYes No Total

Noise Reduction

.11 .19 .30

.56 .14 .70

.67 .33 1.00

Noise Reduction

P N S P N P S P N S( ) ( ) ( ) ( )

70 67 56

.11 .19 .30

.56 .14 .70

.67 .33 1.00

Noise Reduction

P N S( ) . . .

56 14 11

Venn Diagram of the X or Y but not Both Case

P(X or Y but not both) = P(X)+P(Y)-P(X∩Y)-P(X∩Y) = P(XUY) – P(X∩Y)

The Neither/Nor Region

P X Y P X Y( ) ( ) 1

The Neither/Nor Region

P N S P N S( ) ( )

Special Law of Addition

If X and Y are mutually exclusive,

P X Y P X P Y( ) ( ) ( )

Mutually Exclusive• Are mutually exclusive events independent?• A firm interested in increasing its workers

productivity surveyed by asking which one of the following factor needs improving:

Improve safety (S) 21%Improve cafeteria food (F) 28%Improve communication skills(C) 8%Improve supervision(M) 12%Improve room conditions(R) 15%Improve quality of equipments(E) 16%What are P(S U F) and P(S∩F)?

Law 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 X P X

P Y( | )

( | ) ( )

What is the P(S/N)?

• .11 • .19 • .30• .56 • .14 • .70

• .67 • .33 • 1.00

• Increase • Storage Space

• Yes (S) • No • Total• Yes (N)• No• Total

• Noise • Reduc

Law of Conditional Probability

.56 .70

P S NP N S

( | )( )

Example: Raw values

Male Female

Managerial(G) 8 3 11

Professional(P) 31 13 44

Technical(T) 52 17 69

Clerical(C) 9 22 31

100 55 155

A company reveals that 155 employees worked in one of 4 positions as shown below with their gender in columns.

If an employee is selected randomly, what is the probability that the employee is female or a professional worker? P(F/G)? P(T/M)? P(T and M)?

Multiplication Laws

General law of multiplication has to do with joint probability:P(X∩Y) = P(X)P(Y/X) = P(Y) P(X/Y)

Multiplication Rule

Use the following table to develop General Law of Multiplication as well as conditional probability

Male (M) Female(F)

Managerial (G) 0.052 0.019 0.071Professional(P) 0.2 0.084 0.284Technical (T) 0.335 0.110 0.445Clerical(C) 0.058 0.142 0.2

0.645 0.355

What is P(F/G)? 0.019/0.071=0.27 =P(F∩G)/P(G)What is P(T/M)? 0.335/0.645=0.52=P(T∩M)/P(M)P(T∩M) = P(T/M) P(M)

Law of Multiplication Demonstration Problem 4.5

P X Y P X P Y X P Y P X Y( ) ( ) ( | ) ( ) ( | ) Married

Yes No

Supervisor Yes 16 14 30

No 64 46 110

80 60 140

A company has 140 employees of which 30 are supervisors. Eighty are married, and 20% of the married employees are supervisors.

Law of Multiplication Demonstration Problem 4.5

Yes No

.4571 .3286

.1143 .1000 .2143

.5714 .4286 1.00

Married

Supervisor

Probability Matrixof Employees

20.0)|(

5714.0140

2143.0140

P M S P M P S M( ) ( ) ( | )

( . )( . ) .

0 5714 0 20 0 1143

P M S P M P M S

P M S P S P M S

P M P M

( ) ( ) ( )

( ) ( )

0 5714 0 1143 0 4571

0 2143 0 1143 0 1000

1 0 5714 0 4286

P S P S

P M S P S P M S

( ) ( )

( ) ( ) ( )

1 0 2143 0 7857

0 7857 0 4571 0 3286

Special Law of Multiplication for Independent Events

• General Law

• Special LawP 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

( ) ( | ), ( ) ( | ).

( ) ( ) ( )

Office Design Problem

SNPSNP

.19 .30

.14 .70

.33 1.00

Noise Reduction .11

Reduced Sample Space for “Increase

Storage Space” = “Yes”

Independent 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 X and Y are independent events,

, and P X Y P X

P Y X P Y

( | ) ( )

( | ) ( ).

Demonstration problem 4.2

• Probability matrix is given below from a national survey of 200 executives who were asked to identify the geographic location of their company and their company’s industry type.

Independent EventsDemonstration Problem 4.10

Geographic Location

NortheastD

SoutheastE

MidwestF

Finance A .12 .05 .04 .07 .28

Manufacturing B .15 .03 .11 .06 .35

Communications C .14 .09 .06 .08 .37

.41 .17 .21 .21 1.00

P A GP A G

P GP A

P A G P A

( | )( )

.. ( ) .

( | ) . ( ) .

0 210 33 0 28

0 33 0 28

Example : Independence• 115.5 mill civilians of a country’s labour force is

shown below by age and employment status.

• What is P (U|Y)? Is unemployment independent of age? Write joint distribution that will make them independent.

• 13.6%,no

Y (young, under 25)

O (older, 25 and over)

E(employed) 20.4 86.8 107.2

U (unemployed) 3.2 5.1 8.3

23.6 91.9 115.5

E 0.177 0.751 0.928

U 0.028 0.044 0.072

0.205 0.795 1.00

Exercise

• 48% of all bachelor’s degrees are obtained by women & 17.5% of all bachelor’s degrees are in business. 4.7% of all bachelor’s degrees go to women majoring in business. Are the events ‘bachelor’s degree holder is a woman’ and ‘bachelor degrees in business’ statistically significant?

Ans: P(A)P(B)=0.084 =/ P(A & B)• If 90% of all PCs of a particular model will operate for at

least a year before requiring repair. A manager purchases 3 of these computers. What is the probability that all 3 will work for 1 year before any repair is needed?

Ans: 0.729

Only a Bayesian Statistician is able to coherently explain the statement: If there is a 50-50 chance that something can go wrong, then 9 times out of ten it will.

Baye’s Theorem

Total Probability & Bayes’ Theorem

• • Bayes’ theorem allows to evaluate probabilities that are

difficult to obtain• Law of total probability: considering only two events X and Y.• Extension if X can be partitioned into “n” events, Law will

• Ex: If Y= event that a picture card is drawn from a deck. X1 , X2 , X3 , X4 are events that denote card drawn is a heart, diamond, spade , club respectively.

• P(Y) = P(Y∩X1 )+P(Y∩X2 )+P(Y∩X3 )+P(Y∩X4 ) = 16/52

X)P(Y X) P(YP(Y)

i 1i )XP(YP(Y)

Total Probability & Bayes’ Law

• Express total probability in terms of conditional probabilities

• Bayes’ rule is a formula that uses the law of conditional probabilities to allow revision of original probability with arrival of new information.

1ii )P(X)X|P(Y)XP(YP(Y)

...)X|Y(P)X(P)X|Y(P)X(P)X|Y(P)X(P

P(Y)Y)P(X

)Y|P(X2211

Revision of Probabilities: Bayes’ Rule

• 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

n n( | )

( | ) ( )

( | ) ( ) ( | ) ( ) ( | ) ( )

1 1 2 2

Example: Bayes’ Rule

Two companies, Alamo Ribbon and South Jersey produce a type of ribbon. Let Alamo produce 65% and South Jersey 35% of the total market. 8% of the ribbons produced by Alamo are defective whereas 12% of South Jersey’s are defective. A customer purchases a new ribbon. What are the probabilities that Alamo produced the ribbon? South Jersey? The ribbon is tested and found defective. Now what is the probability that Alamo produced the ribbon? That south jersey produced the ribbon?

• Notice the new information that ribbon purchased is defective changes the probability.

Answer:

Out of 100, ‘A’ sales 65 and ‘S’ sales 35.Out of 100, ‘A’ produces 5.2 and ‘S’ produces 4.2 defectivesOut of 100 total defectives= 9.4If 1 defective is observed probability it is due to ‘A’ is

then 5.2/9.4 =0.55If 1 defective is observed probability it is due to ‘B’ is then

4.2/9.4 =0.45

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 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

Revision of Probabilities with Bayes’ Rule: Ribbon Problem

Conditional Probability

0.0520.094

=0.553

0.0420.094

=0.447

South Jersey

Prior Probability

P Ei( )

Joint Probability

P E di( )

Revised Probability

P E di( | )P d Ei( | )

Revision of Probabilities with Bayes' Rule: Ribbon Problem

Alamo0.65

SouthJersey0.35

Defective0.08

Defective0.12

Acceptable0.92

Acceptable0.88

+ 0.094

Homework Exercise• #4.26 (p. 125) #4.33 (p.130) # 4.45 (p.135)

#4.51(p.136) [due on Aug 3]. This is a group assignment for 5-individual group. (submission to be done in clearly hand-written paper.) A randomly chosen individual will present some of the exercises in front of the class)

• Case study for group “Colgate-Palmolive makes a total effort” (this is also due on Aug3; first group from each section -the group of ten- will introduce the case including the answers for presentation in the class. Two randomly chosen students will present the case. Case is in the textbook at the end of chp 4). All works will be assessed.

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