Applied Business Statistics ,ken black , ch 4

Copyright 2010 John Wiley & Sons, Inc. 1
Copyright 2010 John Wiley & Sons, Inc.
Business Statistics, 6th ed.
by Ken Black
Chapter 4
Introduction to
Basic Probability

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.

Probability
Probability – probability of occurrences are
assigned to the inferential process under
conditions of uncertainty

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)

Classical Probability
P E
N
Where
N
en( )
:
=
=
=
total number of outcomes
number of outcomes in Een

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
Classical Probability

Relative Frequency of Occurrence method – the
probability of an event is equal to the number of
times the event has occurred in the past divided by
the total number of opportunities for the event to
have occurred
Frequency of occurrence is based on what has
happened in the past
Relative Frequency Probability

Eproducing
outcomesofnumber
trialsofnumbertotal
:
)(
e
=
=
=
n
n
N
Where
N
EP e
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

Subjective Probability - Comes from a person’s
intuition or reasoning
Subjective -- different individuals may (correctly or
incorrectly) assign different numeric probabilities to
the same event
Degree of belief in the results of the event
Useful for unique (single-trial) experiments
New product introduction
Site selection decisions
Sporting events
Subjective Probability

Experiment – is a process that produces an outcome
Event – an outcome of an experiment
Elementary event – events that cannot be
decomposed or broken down into other events
Sample Space – a complete roster/listing of all
elementary events for an experiment
Trial: one repetition of the process
Structure of Probability

Set Notation – the use of braces to group members
The UNION of x, y is formed by combining elements from
both sets, and is denoted by x U y. Read as “x or y.”
An INTERSECTION is denoted x ∩ y.
The symbol is read as “and”. “x and y”

Mutually Exclusive Events – events such that the
occurrence of one precludes the occurrence of the
other
These events have no intersection
Independent Events – the occurrence or
nonoccurrence of one has no affect on the occurrence
of the others

Collectively Exhaustive Events – listing of all possible
elementary events for an experiment
Complementary Events – two events, one of which
comprises all the elementary events of an
experiment that are not in the other event

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

Family
Children in
Household
Number of
Automobiles
A
B
C
D
Yes
Yes
No
Yes
3
2
1
2
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)
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)
Sample Space: Roster Example

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: Set Notation for
Random Sample of Two Families

 
 
 9,7,6,5,4,3,2,1
6,5,4,3,2
9,7,4,1
=
=
=
YX
Y
X
 
 
 
C IBM DEC Apple
F Apple Grape Lime
C F IBM DEC Apple Grape Lime
=
=
 =
, ,
, ,
, , , ,
YX
The union of two sets contains an instance of each
element of the two sets.
Union of Sets

 
 
 
X
Y
X Y
=
=
 =
1 4 7 9
2 3 4 5 6
4
, , ,
, , , ,
YX
 
 
 
C IBM DEC Apple
F Apple Grape Lime
C F Apple
=
=
 =
, ,
, ,
X  Y
The intersection of two sets contains only those
element common to the two sets.
Intersection of Sets

 
 
 
X
Y
X Y
=
=
 =
1 7 9
2 3 4 5 6
, ,
, , , ,
 
 
 =
=
=
FC
LimeGrapeF
AppleDECIBMC
,
,,
YX
0)( =YXP
Events with no common outcomes
Occurrence of one event precludes the occurrence
of the other event
Mutually Exclusive Events

Independent Events
P X Y P X and P Y X P Y( | ) ( ) ( | ) ( )= =
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.

Collectively Exhaustive Events
E1 E2 E3
Sample Space with three
collectively exhaustive events
Contains all elementary events for an experiment

Complementary Events
Sample
Space
A
P Sample Space( ) = 1
P A P A( ) ( ) = −1
A
All elementary events not in the event ‘A’ are in its
complementary event.

Counting the Possibilities
mn Rule
Sampling from a Population with Replacement
Combinations: Sampling from a Population without
Replacement

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.
How many meals are available?
5 * 4 * 8 * 3 * 4 * 3 = 5760

Combinations: Sampling from a
Population without Replacement
This counting method uses combinations
Selecting n items from a population of N
without replacement

Combinations
Combinations – sampling “n” items from a
population size N without replacement provides the
formula shown below
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
nCr

Combinations: Sampling from a
Population without Replacement
For example, suppose a small law firm has 16
employees and three are to be selected randomly to
represent the company at the annual meeting of the
American Bar Association.
How many different combinations of lawyers could
be sent to the meeting?
Answer: NCn = 16C3 = 16!/(3!13!) = 560.

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
)(XP P X Y( ) P X Y( ) P X Y( | )

General Law of Addition
P X Y P X P Y P X Y( ) ( ) ( ) ( ) = + − 
YX

General Law of Addition -- Example
)()()()( SNPSPNPSNP −+=
81.0
56.67.70.)(
56.)(
67.)(
70.)(
=
−+=
=
=
=
SNP
SNP
SP
NPSN
.56
.67.70

81.
56.67.70.
)()()()(
=
−+=
−+= SNPSPNPSNP
.11 .19 .30
.56 .14 .70
.67 .33 1.00
Increase
Storage Space
Yes No Total
Yes
No
Total
Noise
Reduction
Office Design Problem Probability Matrix

If a worker is randomly selected from the company
described in Demonstration Problem 4.1, what is the
probability that the worker is either technical or
clerical? What is the probability that the worker is
either a professional or a clerical?
Demonstration problem 4.3

Examine the raw value matrix of the company’s human
resources data shown in Demonstration Problem 4.1. In
many raw value and probability matrices like this one, the
rows are non-overlapping or mutually exclusive, as are
the columns. In this matrix, a worker can be classified as
being in only one type of position and as either male or
female but not both. Thus, the categories of type of
position are mutually exclusive, as are the categories of
sex, and the special law of addition can be applied to the
human resource data to determine the union
probabilities.

Let T denote technical, C denote clerical, and P denote
professional. The probability that a worker is either
technical or clerical is
P(T U C) = P (T) + P (C) = 69/155 + 31/155 = 100/155 = .645
The probability that a worker is either professional or
clerical is
P (P U C) = P (P) + P (C) = 44/155 + 31/155 = 75/155 = .484

P T C P T P C( ) ( ) ( )
.
 = +
= +
=
69
155
31
155
645
Demonstration Problem 4.3
Type of Gender
Position Male Female Total
Managerial 8 3 11
Professional 31 13 44
Technical 52 17 69
Clerical 9 22 31
Total 100 55 155

Type of Gender
Position Male Female Total
Managerial 8 3 11
Professional 31 13 44
Technical 52 17 69
Clerical 9 22 31
Total 100 55 155
P P C P P P C( ) ( ) ( )
.
 = +
= +
=
44
155
31
155
484

)|()()|()()( YXPYPXYPXPYXP ==
P M
P S M
P M S P M P S M
( ) .
( | ) .
( ) ( ) ( | )
( . )( . ) .
= =
=
 = 
= =
80
140
0 5714
0 20
0 5714 0 20 0 1143
Law of Multiplication

The intersection of two events is called the joint
probability
General law of multiplication is used to find the joint
probability
General law of multiplication gives the probability
that both events x and y will occur at the same time
P(x|y) is a conditional probability that can be stated
as the probability of x given y

If a probability matrix is constructed for a problem,
the easiest way to solve for the joint probability is to
find the appropriate cell in the matrix and select the
answer

Total
.7857
Yes No
.4571 .3286
.1143 .1000 .2143
.5714 .4286 1.00
Married
Yes
No
Total
Supervisor
Probability Matrix
of Employees
20.0)|(
5714.0
140
80
)(
2143.0
140
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
3286.04571.07857.0
)()()(
7857.02143.01
)(1)(
=−=
−=
=−=
−=
SMPSPSMP
SPSP

Law of Conditional Probability
Conditional probability are based on the prior
knowledge you have on one of the two events being
studied
If X and Y are two events, the conditional probability
of X occurring given that Y is known or has occurred
is expressed as P(X|Y)

)(
)()|(
)(
)(
)|(
YP
XPXYP
YP
YXP
YXP

=

=
The conditional probability of X given Y is the joint
probability of X and Y divided by the marginal
probability of Y.

70% of respondents believe noise reduction would
improve productivity.
56% of respondents believed both noise reduction
and increased storage space would improve
productivity
A worker is selected randomly and asked about
changes in the office design
What is the probability that a randomly selected
person believes storage space would improve
productivity given that the person believes noise
reduction improves productivity?

P N
P N S
P S N
P N S
P N
( ) .
( ) .
( | )
( )
( )
.
.
.
=
 =
=

=
=
70
56
56
70
80
NS
.56
.70

Independent Events
When X and Y are independent, the conditional
probability is solved as a marginal probability

Geographic Location
Northeast
D
Southeast
E
Midwest
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
P A G
P G
P A
P A G P A
( | )
( )
( )
.
.
. ( ) .
( | ) . ( ) .
=

= = =
=  =
007
021
033 028
033 028
Independent Events

Revision of Probabilities: Bayes’ Rule
)()|()()|()()|(
)()|(
)|(
2211 nn
ii
i
XPXYPXPXYPXPXYP
XPXYP
YXP
++
=
An extension to the conditional law of probabilities
Enables revision of original probabilities with new
information

Bayes’ Rule
Bays’ rule – extends the use of the law of conditional
probabilities to all revision of original probabilities
with new information

Bayes’ Rule
Note, the numerator of Bayes’ Rule and the law of
conditional probability are the same
The denominator is a collective exhaustive listing of
mutually exclusive outcomes of Y
The denominator is a weighted average of the conditional
probabilities with the weights being the prior probabilities
of the corresponding event

Revision of Probabilities
with Bayes’ Rule: Ribbon Problem
P Alamo
P SouthJersey
P d Alamo
P d SouthJersey
P Alamo d
P d Alamo P Alamo
P d Alamo P Alamo P d SouthJersey P SouthJersey
P SouthJerseyd
P 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 35
0 553
0 12 0 35
0 08)( . ) ( . )( . )
.
0 65 0 12 0 35
0 447
+
=

Conditional
Probability
0.052
0.042
0.094
0.65
0.35
0.08
0.12
0.052
0.094
=0.553
0.042
0.094
=0.447
Alamo
South Jersey
Event
Prior
Probability
P Ei( )
Joint
Probability
P E di( )
Revised
Probability
P E di( | )P d Ei( | )

Alamo
0.65
South
Jersey
0.35
Defective
0.08
Defective
0.12
Acceptable
0.92
Acceptable
0.88
0.052
0.042
+ 0.094

Applied Business Statistics ,ken black , ch 4

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