ch04.probability
TRANSCRIPT
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1
Business Statistics, 4eby Ken Black
Chapter 4Probability
A LOTTERY IS A TAX ON PEOPLE
WHO DON'T UNDERSTAND STATISTICS!!
Discrete Distributions
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-2
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.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-3
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)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-4
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
P EN
Where
N
en( )
:
total number of outcomes
number of outcomes in Een
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-5
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
P EN
Where
N
en( )
:
total number of trials
number of outcomes
producing Een
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-6
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
4-7
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-8
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,
E1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-9
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
ABCD
YesYesNoYes
3212
Tiny Town Population
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-10
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-11
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
ABCD
YesYesNo
Yes
3212
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)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-12
Sample Space: Tree Diagram for Random Sample of Two Families
A
B
C
D
D
BC
D
A
C
D
A
B
C
A
B
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-13
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-14
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-15
Union of Sets• The union of two sets contains an instance
of each element of the two sets.
X
Y
X Y
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
, ,
, ,
, , , ,
YX
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-16
Intersection of Sets• The intersection of two sets contains only
those element common to the two sets.
X
Y
X Y
1 4 7 9
2 3 4 5 6
4
, , ,
, , , ,
C IBM DEC Apple
F Apple Grape Lime
C F Apple
, ,
, ,
YX
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-17
Mutually Exclusive Events
• Events with no common outcomes
• Occurrence of one event precludes the occurrence of the other event
X
Y
X Y
1 7 9
2 3 4 5 6
, ,
, , , ,
C IBM DEC Apple
F Grape Lime
C F
, ,
,
YX
P X Y( ) 0
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-18
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.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-19
Collectively Exhaustive Events
• Contains all elementary events for an experiment
E1 E2 E3
Sample Space with three collectively exhaustive events
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-20
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-21
Four Types of Probability
• Marginal Probability• Union Probability• Joint Probability• Conditional Probability
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-22
Four 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( | )
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-23
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( ) ( ) ( ) ( )
YX
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?
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-24
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-25
General 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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-26
Office Design ProblemProbability Matrix
.11 .19 .30
.56 .14 .70
.67 .33 1.00
Increase Storage SpaceYes No Total
Yes
No
Total
Noise Reduction
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-27
Office Design ProblemProbability Matrix
.11 .19 .30
.56 .14 .70
.67 .33 1.00
Increase Storage SpaceYes No Total
Yes
No
Total
Noise Reduction
P N S P N P S P N S( ) ( ) ( ) ( )
. . .
.
70 67 56
81
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-28
Office Design ProblemProbability Matrix
.11 .19 .30
.56 .14 .70
.67 .33 1.00
Increase Storage SpaceYes No Total
Yes
No
Total
Noise Reduction
P N S( ) . . .
.
56 14 11
81
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-29
Venn Diagram of the X or Y but not Both Case
YX
P(X or Y but not both) = P(X)+P(Y)-P(X∩Y)-P(X∩Y) = P(XUY) – P(X∩Y)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-30
The Neither/Nor Region
YX
P X Y P X Y( ) ( ) 1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-31
The Neither/Nor Region
SN
P N S P N S( ) ( )
.
.
1
1 81
19
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-32
Special Law of Addition
If X and Y are mutually exclusive,
P X Y P X P Y( ) ( ) ( )
X
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)?
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-33
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-34
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
P Y X P X
P Y( | )
( )
( )
( | ) ( )
( )
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-35
Office Design ProblemProbability Matrix
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
tion
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-36
Law of Conditional Probability
NS
.56 .70
P N
P N S
P S NP N S
P N
( ) .
( ) .
( | )( )
( )
.
..
70
56
56
7080
Example: Raw values
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-37
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-38
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-39
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)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-40
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.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-41
Law of Multiplication Demonstration Problem 4.5
Total
.7857
Yes No
.4571 .3286
.1143 .1000 .2143
.5714 .4286 1.00
Married
YesNo
Total
Supervisor
Probability Matrixof Employees
20.0)|(
5714.0140
80)(
2143.0140
30)(
MSP
MP
SP
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
1 0 5714 0 4286
P S P S
P M S P S P M S
( ) ( )
. .
( ) ( ) ( )
. . .
1
1 0 2143 0 7857
0 7857 0 4571 0 3286
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-42
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
( ) ( | ), ( ) ( | ).
,
( ) ( ) ( )
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-43
Office Design Problem
164.
67.
11.
)(
)()|(
SP
SNPSNP
.19 .30
.14 .70
.33 1.00
Increase Storage SpaceYes No Total
YesNo
Total
Noise Reduction .11
.56
.67
Reduced Sample Space for “Increase
Storage Space” = “Yes”
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-44
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.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-45
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-46
Independent EventsDemonstration Problem 4.10
Geographic Location
NortheastD
SoutheastE
MidwestF
WestG
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 07
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-47
Age
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-48
Y O
E 0.177 0.751 0.928
U 0.028 0.044 0.072
0.205 0.795 1.00
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-49
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-50
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-51
• • 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
be:
• 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)
n
i 1i )XP(YP(Y)
Total Probability & Bayes’ Law
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-52
• 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.
n
1iii
n
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
iiii
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-53
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
i i
n n( | )
( | ) ( )
( | ) ( ) ( | ) ( ) ( | ) ( )
1 1 2 2
Example: Bayes’ Rule
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-54
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.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-55
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-56
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-57
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( | )
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-58
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
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.
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-59