Download Chapter 4

Business Statistics, 4e
by Ken Black
Discrete Distributions
Chapter 4
Probability
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-1
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.
4-2
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.
4-3
Classical Probability
• Number of outcomes leading
n
to the event divided by the
P( E )  e
total number of outcomes
N
possible
Where:
• Each outcome is equally likely
N  total number of outcomes
• Determined a priori -- before
performing the experiment
ne  number of outcomes in E
• Applicable to games of chance
• Objective -- everyone correctly
using the method assigns an
identical probability
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-4
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
P( E ) 
n
e
N
Where:
N  total number of trials
n
e
 number of outcomes
producing E
4-5
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.
4-6
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.
4-7
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.
4-8
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
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Tiny Town Population
Family
Children in
Household
Number of
Automobiles
A
B
C
D
Yes
Yes
No
Yes
3
2
1
2
4-9
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.
4-10
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
A
B
C
D
Children in
Household
Number of
Automobiles
Yes
Yes
No
Yes
3
2
1
2
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
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)
4-11
Sample Space: Tree Diagram for
Random Sample of Two Families
A
B
C
D
A
B
C
D
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
C
D
A
B
D
A
B
C
4-12
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.
4-13
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)
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Venn Diagram
4-14
Union of Sets
• The union of two sets contains an instance
of each element of the two sets.
X  1,4,7,9
Y  2,3,4,5,6
X
Y
X  Y  1,2,3,4,5,6,7,9
C   IBM , DEC , Apple
F   Apple, Grape, Lime
C  F   IBM , DEC , Apple, Grape, Lime
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-15
Intersection of Sets
• The intersection of two sets contains only
those element common to the two sets.
X  1,4,7,9
Y  2,3,4,5,6
X
Y
X  Y   4
C   IBM , DEC , Apple
F   Apple, Grape, Lime
C  F   Apple
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-16
Mutually Exclusive Events
• Events with no
common outcomes
• Occurrence of one
event precludes the
occurrence of the
other event
C   IBM , DEC , Apple
F   Grape, Lime
CF 

Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
X
X  1,7,9
Y  2,3,4,5,6
X Y  
Y
P( X  Y )  0

4-17
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.
4-18
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.
4-19
Complementary Events
• All elementary events not in the event ‘A’
are in its complementary event.
Sample
Space
A
A
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
P(Sample Space) 1
P( A)  1  P( A)
4-20
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.
4-21
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.
4-22
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.
4-23
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?
N!
1000!
N

 166,167,000
 
 n  n!( N  n)! 3!(1000  3)!
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-24
Four Types of Probability
•
•
•
•
Marginal Probability
Union Probability
Joint Probability
Conditional Probability
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-25
Four Types of Probability
Marginal
Union
Joint
Conditional
P( X )
P( X  Y )
P( X  Y )
P( X | Y )
The probability
of X occurring
X
The probability
of X or Y
occurring
X Y
The probability
of X and Y
occurring
The probability
of X occurring
given that Y
has occurred
X Y
Y
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-26
General Law of Addition
P( X  Y )  P( X)  P(Y )  P( X  Y )
X
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Y
4-27
General Law of Addition -- Example
P( N  S )  P( N )  P( S )  P( N  S )
P ( N ) .70
S
N
P ( S ) .67
.70
.56
.67
P ( N  S ) .56
P ( N  S ) .70.67 .56
 0.81
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-28
Office Design Problem
Probability Matrix
Noise
Reduction
Yes
No
Total
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Increase
Storage Space
Yes
No
.14
.56
.19
.11
.33
.67
Total
.70
.30
1.00
4-29
Office Design Problem
Probability Matrix
Noise
Reduction
Yes
No
Total
Increase
Storage Space
Yes
No
.14
.56
.19
.11
.33
.67
Total
.70
.30
1.00
P( N  S )  P( N )  P( S )  P( N  S )
.70.67 .56
.81
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-30
Office Design Problem
Probability Matrix
Noise
Reduction
Yes
No
Total
Increase
Storage Space
Yes
No
.14
.56
.19
.11
.33
.67
Total
.70
.30
1.00
P( N  S ) .56.14 .11
.81
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-31
Venn Diagram of the X or Y
but not Both Case
X
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Y
4-32
The Neither/Nor Region
X
Y
P( X  Y )  1  P( X  Y )
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-33
The Neither/Nor Region
N
S
P( N  S )  1  P( N  S )
 1.81
.19
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-34
Special Law of Addition
If X and Y are mutually exclusive,
P( X  Y )  P( X )  P(Y )
Y
X
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-35
Demonstration Problem 4.3
Type of
Position
Managerial
Professional
Technical
Clerical
Total
Gender
Male Female
8
3
31
13
52
17
9
22
100
55
Total
11
44
69
31
155
P(T  C)  P(T )  P(C)
69 31


155 155
.645
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-36
Demonstration Problem 4.3
Type of
Position
Managerial
Professional
Technical
Clerical
Total
Gender
Male Female
8
3
31
13
52
17
9
22
100
55
Total
11
44
69
31
155
P( P  C)  P( P)  P(C)
44
31


155 155
.484
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-37
Law of Multiplication
Demonstration Problem 4.5
P( X  Y )  P( X)  P(Y| X)  P(Y )  P( X| Y )
80
P( M ) 
 0. 5714
140
P( S| M )  0. 20
P ( M  S )  P ( M )  P ( S| M )
 ( 0. 5714 )( 0. 20 )  0.1143
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-38
Law of Multiplication
Demonstration Problem 4.5
Probability Matrix
of Employees
Supervisor
Yes
No
Total
Married
Yes
No
Total
.1143 .1000 .2143
.4571 .3286 .7857
.5714 .4286 1.00
30
 0.2143
140
80
P( M ) 
 0.5714
140
P ( S | M )  0.20
P(S ) 
P( M  S)  P( M )  P( S| M )
 (0. 5714)(0. 20)  0.1143
P( M  S )  P( M )  P( M  S )
P( S )  1  P( S )
 0. 5714  0.1143  0. 4571
 1  0. 2143  0. 7857
P( M  S )  P( S )  P( M  S )
 0. 2143  0.1143  0.1000
P( M  S )  P( S )  P( M  S )
 0. 7857  0. 4571  0. 3286
P( M )  1  P( M )
 1  0. 5714  0. 4286
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-39
Special Law of Multiplication
for Independent Events
• General Law
P( X  Y )  P( X)  P(Y| X)  P(Y )  P( X| Y )
• Special Law
If events X and Y are independent ,
P( X )  P( X | Y ), and P(Y )  P(Y | X ).
Consequently,
P( X  Y )  P( X )  P(Y )
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-40
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  Y ) P(Y | X )  P( X )
P( X| Y ) 

P(Y )
P(Y )
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-41
Law of Conditional Probability
P ( N ) .70
P ( N  S ) .56
N
S
.56
.70
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
P( N  S )
P( S | N ) 
P( N )
.56

.70
.80
4-42
Office Design Problem
Noise
Reduction
Yes
No
Total
Reduced Sample
Space for
“Increase
Storage Space”
= “Yes”
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
Increase
Storage Space
Yes
No
.14
.56
.19
.11
.33
.67
Total
.70
.30
1.00
P( N  S ) .11
P( N | S ) 

P( S )
.67
 .164
4-43
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 ,
P( X | Y )  P( X ), and
P(Y | X )  P(Y ).
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-44
Independent Events
Demonstration Problem 4.10
Geographic Location
Northeast Southeast Midwest
D
E
F
West
G
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  G ) 0.07
P( A| G ) 

 0.33
P(G )
0.21
P( A| G )  0.33  P( A)  0.28
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
P( A)  0.28
4-45
Independent Events
Demonstration Problem 4.11
D
E
A
8
12
20
B
20
30
50
C
6
9
15
34
51
85
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
8
P ( A| D) 
.2353
34
20
P ( A) 
.2353
85
P ( A| D)  P ( A)  0.2353
4-46
Revision of Probabilities: Bayes’ Rule
• An extension to the conditional law of
probabilities
• Enables revision of original probabilities
with new information
P( Xi| Y ) 
P(Y | Xi ) P( Xi )
P(Y | X 1) P( X 1)  P(Y | X 2 ) P( X 2 )  P(Y | Xn ) P( Xn )
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-47
Revision of Probabilities
with Bayes' Rule: Ribbon Problem
P( Alamo)  0. 65
P( SouthJersey)  0. 35
P( d | Alamo)  0. 08
P( d | SouthJersey)  0.12
P( d | Alamo)  P( Alamo)
P( d | Alamo)  P( Alamo)  P( d | SouthJersey)  P( SouthJersey)
( 0. 08)( 0. 65)

 0. 553
( 0. 08)( 0. 65)  ( 0.12 )( 0. 35)
P( d | SouthJersey)  P( SouthJersey)
P( SouthJersey| d ) 
P( d | Alamo)  P( Alamo)  P( d | SouthJersey)  P( SouthJersey)
( 0.12 )( 0. 35)

 0. 447
( 0. 08)( 0. 65)  ( 0.12 )( 0. 35)
P( Alamo| d ) 
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-48
Revision of Probabilities
with Bayes’ Rule: Ribbon Problem
Prior
Probability
Event
Alamo
P ( Ei )
0.65
Conditional
Probability
Joint
Probability
Revised
Probability
P(d| Ei )
P(Ei  d) P( Ei| d )
0.08
0.052
0.052
0.094
=0.553
South Jersey
0.35
0.12
0.042
0.042
0.094
0.094
=0.447
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-49
Revision of Probabilities
with Bayes' Rule: Ribbon Problem
Defective
0.08
0.052
Alamo
0.65
+
Acceptable
0.92
Defective
0.12
South
Jersey
0.35
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
0.094
0.042
Acceptable
0.88
4-50
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.
4-51
Probability for a Sequence
of Independent Trials
• Probability of specific sequences containing
1 commercial and 2 retail customers,
assuming the events C and R are
independent
 1   3  3 9
P(C1  R 2  R 3)  P(C ) P( R) P( R)        
 4   4   4  64
 3  1  3 9
P( R1  C 2  R 3)  P( R) P(C ) P( R)        
 4   4   4  64
 3  3  1  9
P( R1  R 2  C 3)  P( R) P( R) P(C )        
 4   4   4  64
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-52
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(C1  R 2  R 3)  ( R1  C 2  R 3)  ( R1  R 2  C 3)
 P(C1  R 2  R 3)  P( R1  C 2  R 3)  P( R1  R 2  C 3)
9 9 9 27
   
64 64 64 64
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-53
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
9


P C  R  R  P(C )  P( R)  P( R) 
64
• Number of sequences containing 1 commercial and 2
retail customers
 n
n!
3!
nCr    

3
 r  r ! n  r  ! 1! 3  1 !
• Probability of a sequence containing 1 commercial and 2
retail customers
 3 
9  27

 64  64
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-54
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 returns
E1 E2 N 3 N 4
E1 N 2 E3 N 4
E1 N 2 N 3 E4
N1 E2 E3 N4
N1 E2 N 3 E4
N1 N2 E3 E4
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-55
Probability for a Sequence
of Dependent Trials
• Probability of specific sequences containing 2
erroneous tax returns (three of the six
sequences)
P( E 1  E 2  N 3  N 4)  P ( E 1) P ( E 2| E 1) P( N 3| E 1  E 2 ) P ( N 4| E 1  E 2  N 3)
55,552
 8   7   32   31 
      
 0.01
 50   49   48   47  5,527,200
P( E 1  N 2  E 3  N 4)  P ( E 1) P ( N 2| E 1) P( E 3| E 1  N 2 ) P ( N 4| E 1  N 2  E 3)
55,552
 8   32   7   31 
      
 0.01
 50   49   48   47  5,527,200
P( E 1  N 2  N 3  E 4)  P ( E 1) P ( N 2| E 1) P( N 3| E 1  N 2 ) P ( E 4| E 1  N 2  N 3)
55,552
 8   32   31  7 
      
 0.01
 50   49   48   47  5,527,200
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-56
Probability for a Sequence
of Independent Trials
• Probability of observing a sequence containing
exactly 2 erroneous tax returns
P(( E1  E 2  N 3  N 4 )  ( E1  N 2  E 3  N 4 )  ( E1  N 2  N 3  E 4 )
( N 1  E 2  E 3  N 4 )  ( N 1  E 2  N 3  E 4 )  ( N 1  N 2  E 3  E 4 ))
 P( E 1  E 2  N 3  N 4 )  P( E 1  N 2  E 3  N 4 )  P( E 1  N 2  N 3  E 4 )
 P( N 1  E 2  E 3  N 4 )  P( N 1  E 2  N 3  E 4 )  P( N 1  N 2  E 3  E 4 )
55, 552
55, 552
55, 552
55, 552
55, 552
55, 552






5, 527, 200 5, 527, 200 5, 527, 200 5, 527, 200 5, 527, 200 5, 527, 200
 0. 06
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-57
Probability for a Sequence
of Dependent Trials
• Probability of a specific sequence with exactly 2 erroneous tax
returns
55,552
 8   7   32   31 
P( E 1  E 2  N 3  N 4)    
 0.01


 
 50   49   48   47  5,527,200
• Number of sequences containing exactly 2 erroneous tax
returns
 n
n!
4!
n
nCr     C 

6
r
r ! n  r  ! 2! 4  2 !
 r
• Probability of a sequence containing exactly 2 erroneous tax
returns
 55,552 
  0.06
 5,527,200 
 6 
Business Statistics, 4e, by Ken Black. © 2003 John Wiley & Sons.
4-58