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Transcript of Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken...
![Page 1: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/1.jpg)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1
Business Statistics, 4eby Ken Black
Chapter 4
Probability
Discrete Distributions
![Page 2: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/2.jpg)
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.
![Page 3: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/3.jpg)
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)
![Page 4: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/4.jpg)
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
![Page 5: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/5.jpg)
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
![Page 6: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/6.jpg)
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
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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
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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
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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
![Page 10: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/10.jpg)
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
![Page 11: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/11.jpg)
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)
![Page 12: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/12.jpg)
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
![Page 13: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/13.jpg)
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
![Page 14: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/14.jpg)
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
![Page 15: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/15.jpg)
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
![Page 16: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/16.jpg)
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
![Page 17: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/17.jpg)
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
![Page 18: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/18.jpg)
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( | ) ( ) ( | ) ( )
![Page 19: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/19.jpg)
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
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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
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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
![Page 22: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/22.jpg)
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?
![Page 23: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/23.jpg)
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
![Page 24: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/24.jpg)
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
![Page 25: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/25.jpg)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-25
Four Types of Probability
• Marginal Probability• Union Probability• Joint Probability• Conditional Probability
![Page 26: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/26.jpg)
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( | )
![Page 27: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/27.jpg)
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
![Page 28: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/28.jpg)
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
![Page 29: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/29.jpg)
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
![Page 30: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/30.jpg)
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
![Page 31: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/31.jpg)
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
![Page 32: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/32.jpg)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-32
Venn Diagram of the X or Y but not Both Case
YX
![Page 33: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/33.jpg)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-33
The Neither/Nor Region
YX
P X Y P X Y( ) ( ) 1
![Page 34: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/34.jpg)
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
![Page 35: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/35.jpg)
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
![Page 36: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/36.jpg)
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
![Page 37: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/37.jpg)
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
![Page 38: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/38.jpg)
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
![Page 39: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/39.jpg)
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
![Page 40: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/40.jpg)
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
( ) ( | ), ( ) ( | ).
,
( ) ( ) ( )
![Page 41: Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons. 4-1 Business Statistics, 4e by Ken Black Chapter 4 Probability.](https://reader036.fdocuments.us/reader036/viewer/2022082215/56649e9f5503460f94ba1796/html5/thumbnails/41.jpg)
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( | )
( )
( )
( | ) ( )
( )
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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
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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”
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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
( | ) ( )
( | ) ( ).
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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
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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
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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
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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
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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( | )
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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
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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
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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
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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
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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
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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
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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
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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
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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