business statistics, 4e, by ken black. © 2003 john wiley & sons. 4-1 business statistics, 4e by...
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Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1
Business Statistics, 4eby Ken Black
Chapter 4
Probability
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
• 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
• Number of outcomes leading to the event divided by the total number of outcomes possible
• Each outcome is equally likely• Determined a priori -- before
performing the experiment• 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
(correctly) assign different numeric probabilities to the same event
• Degree of belief• Useful for unique (single-trial) experiments
– New product introduction– Initial public offering of common stock– Site selection decisions– Sporting events
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-7
Structure of Probability
• Experiment• Event• Elementary Events• Sample Space• 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– Only one outcome per trial
• Trial: one repetition of the process• Elementary Event: cannot be decomposed or
broken down into other events• 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.
P X Y P X and P Y X P Y( | ) ( ) ( | ) ( )
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
Counting the Possibilities
• mn Rule• Sampling from a Population with
Replacement• Combinations: Sampling from a Population
without Replacement
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-22
mn Rule
• If an operation can be done m ways and a second operation can be done n ways, then there are mn ways for the two operations to occur in order.
• A cafeteria offers 5 salads, 4 meats, 8 vegetables, 3 breads, 4 desserts, and 3 drinks. A meal is two servings of vegetables, which may be identical, and one serving each of the other items. How many meals are available?
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-23
Sampling from a Population with Replacement
• A tray contains 1,000 individual tax returns. If 3 returns are randomly selected with replacement from the tray, how many possible samples are there?
• (N)n = (1,000)3 = 1,000,000,000
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-24
Combinations
• A tray contains 1,000 individual tax returns. If 3 returns are randomly selected without replacement from the tray, how many possible samples are there?
0166,167,00)!31000(!3
!1000
)!(!
!
nNn
N
n
N
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-25
Four Types of Probability
• Marginal Probability• Union Probability• Joint Probability• Conditional Probability
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-26
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. 4-27
General Law of Addition
P X Y P X P Y P X Y( ) ( ) ( ) ( )
YX
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-28
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. 4-29
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. 4-30
Office Design ProblemProbability Matrix
.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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-31
Office Design ProblemProbability Matrix
.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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-32
Venn Diagram of the X or Y but not Both Case
YX
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-33
The Neither/Nor Region
YX
P X Y P X Y( ) ( ) 1
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-34
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. 4-35
Special Law of Addition
If X and Y are mutually exclusive,
P X Y P X P Y( ) ( ) ( )
X
Y
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-36
Demonstration 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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-37
Demonstration 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 P C P P P C( ) ( ) ( )
.
44
155
31
155484
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-38
Law of Multiplication 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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-39
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. 4-40
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. 4-41
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. 4-42
Law of Conditional Probability
NS
.56 .70
P N
P N S
P S NP N S
P N
( ) .
( ) .
( | )( )
( )
.
..
70
56
56
7080
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
( | ) ( )
( | ) ( ).
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-45
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-46
Independent EventsDemonstration Problem 4.11
D E
A 8 12 20
B 20 30 50
C 6 9 15
34 51 85
P A D
P A
P A D P A
( | ) .
( ) .
( | ) ( ) .
8
342353
20
852353
0 2353
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-47
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-48
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. 4-49
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. 4-50
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-51
Probability for a Sequence of Independent Trials
• 25 percent of a bank’s customers are commercial (C) and 75 percent are retail (R).
• Experiment: Record the category (C or R) for each of the next three customers arriving at the bank.
• Sequences with 1 commercial and 2 retail customers.– C1 R2 R3
– R1 C2 R3
– R1 R2 C3
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-52
Probability for a Sequenceof Independent Trials
• Probability of specific sequences containing 1 commercial and 2 retail customers, assuming the events C and R are independent
P C R R P C P R P R
P R C R P R P C P R
P R R C P R P R P C
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
1 2 3
1 2 3
1 2 3
1
4
3
4
3
4
9
64
3
4
1
4
3
4
9
64
3
4
3
4
1
4
9
64
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-53
Probability for a Sequence of Independent Trials
• Probability of observing a sequence containing 1 commercial and 2 retail customers, assuming the events C and R are independent
P C R R R C R R R C
P C R R P R C R P R R C
( ) ( ) ( )
( ) ( ) ( )
1 2 3 1 2 3 1 2 3
1 2 3 1 2 3 1 2 3
9
64
9
64
9
64
27
64
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-54
Probability for a Sequence of Independent Trials
• Probability of a specific sequence with 1 commercial and 2 retail customers, assuming the events C and R are independent
• Number of sequences containing 1 commercial and 2 retail customers
• Probability of a sequence containing 1 commercial and 2 retail customers
P C R R P C P R P R ( ) ( ) ( )9
64
n rCn
rn
r n r
!
! !
!
! !
3
1 3 13
39
64
27
64
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-55
Probability for a Sequence of Dependent Trials
• Twenty percent of a batch of 40 tax returns contain errors.
• Experiment: Randomly select 4 of the 40 tax returns and record whether each return contains an error (E) or not (N).
• Outcomes with exactly 2 erroneous tax returnsE1 E2 N3 N4
E1 N2 E3 N4
E1 N2 N3 E4
N1 E2 E3 N4
N1 E2 N3 E4
N1 N2 E3 E4
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-56
Probability for a Sequence of Dependent Trials
• Probability of specific sequences containing 2 erroneous tax returns (three of the six sequences)
P E E N N P E P E E P N E E P N E E N
P E N E N P E P N E P E E N P N E N E
( ) ( ) ( | ) ( | ) ( | )
,
, ,.
( ) ( ) ( | ) ( | ) ( | )
1 2 3 4 1 2 1 3 1 2 4 1 2 3
1 2 3 4 1 2 1 3 1 2 4 1 2 3
8
50
7
49
32
48
31
47
55 552
5 527 2000 01
8
50
32
49
7
48
31
47
55 552
5 527 2000 01
8
50
32
49
31
48
7
47
55 552
5 527 2000 01
1 2 3 4 1 2 1 3 1 2 4 1 2 3
,
, ,.
( ) ( ) ( | ) ( | ) ( | )
,
, ,.
P E N N E P E P N E P N E N P E E N N
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-57
Probability for a Sequence of Independent Trials
• Probability of observing a sequence containing exactly 2 erroneous tax returns
P E E N N E N E N E N N E
N E E N N E N E N N E E
P E E N N P E N E N P E N N E
P N E E N P N E N E P N N E E
(( ) ( ) ( )
( ) ( ) ( ))
( ) ( ) ( )
( ) ( ) ( )
,
1 2 3 4 1 2 3 4 1 2 3 4
1 2 3 4 1 2 3 4 1 2 3 4
1 2 3 4 1 2 3 4 1 2 3 4
1 2 3 4 1 2 3 4 1 2 3 4
55 552
5
, ,
,
, ,
,
, ,
,
, ,
,
, ,
,
, ,
.
527 200
55 552
5 527 200
55 552
5 527 200
55 552
5 527 200
55 552
5 527 200
55 552
5 527 200
0 06
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-58
Probability for a Sequence of Dependent Trials
• Probability of a specific sequence with exactly 2 erroneous tax returns
• Number of sequences containing exactly 2 erroneous tax returns
• Probability of a sequence containing exactly 2 erroneous tax returns
n r r
nC
n
rn
r n rC
!
! !
!
! !
4
2 4 26
655 552
5 527 2000 06
,
, ,.
P E E N N( ),
, ,.1 2 3 4
8
50
7
49
32
48
31
47
55 552
5 527 2000 01