Chapter 4
Probability
Probability Defined
A probability is a number between 0 and 1 that
measures the chance or likelihood that some
event or set of events will occur.
Assigning Basic Probabilities
• Classical Approach
• Relative Frequency Approach
• Subjective Approach
Classical Approach
F
P(A) =
T
where P(A) = probability of event A
F = number of outcomes “favorable” to event A
T = total number of outcomes possible in the experiment
Relative Frequency Approach
n
P(A) =
N
where N = total number of observations
or trials
n = number of times that event A
occurs
The Language of Probability
•
•
•
•
•
•
Simple Probability
Conditional Probability
Independent Events
Joint Probability
Mutually Exclusive Events
Either/Or Probability
Statistical Independence
Two events are said to be statistically
independent if the occurrence of one
event has no influence on the likelihood of
occurrence of the other.
Statistical Independence
P(A l B) = P(A)
“given”
and
(4.1)
P (B l A) = P(B)
“given”
General Multiplication Rule
P(A  B) = P(A)∙ P(BlA)
“and”
(4.2)
Multiplication Rule for
Independent Events
P(A  B) = P(A) ∙P(B)
(4.3)
Mutually Exclusive Events
Two events, A and B, are said to be
mutually exclusive if the occurrence of
one event means that the other event
cannot or will not occur.
Mutually Exclusive Events
P(A  B) = 0
(4.4)
General Addition Rule
(4.5)
P(A  B) = P(A) + P(B) - P(A  B)
“or”
Addition Rule for
Mutually Exclusive Events
P(A  B) = P(A) + P(B)
(4.6)
“Conditional Equals JOINT Over
SIMPLE” Rule
P( A  B)
P(B I A) =
P( A)
P(A I B) = P ( A  B )
P( B)
(4.7a)
(4.7b)
Complementary Events Rule
P(A′ ) = 1 – P(A)
(4.8)
Figure 4.1
Venn Diagram for the
Internet Shoppers Example
Sample Space (1.0)
A(.8)
B(.6)
(Airline
Ticket
Purchase)
(Book
Purchase)
A∩ B
(.5)
The sample space contains 100% of the possible outcomes in the
experiment. 80% of these outcomes are in Circle A; 60% are in
Circle B; 50% are in both circles.
Figure 4.2
Complementary Events
Sample Space (1.0)
A
A’ (.2)
.8
(everything in the
sample space
outside A)
The events A and A’ are said to be complementary since one or the other (but
never both) must occur. For such events, P(A’ ) = 1 - P(A).
Figure 4.3
Mutually Exclusive Events
Sample Space (1.0)
A
B
Mutually exclusive events appear as non-overlapping circles in a Venn diagram.
Figure 4.4
Probability Tree for the
Project Example
STAGE 1
PROJECT A
PERFORMANCE
STAGE 2
PROJECT B
PERFORMANCE
B
A
A is under budget
B is under budget
B'
B is not under budget
B
A'
A is not under budget
B is under budget
B'
B is not under budget
Figure 4.5
Showing Probabilities
on the Tree
STAGE 1
PROJECT A
PERFORMANCE
STAGE 2
PROJECT B
PERFORMANCE
B(.6)
A (.25)
A is under budget
B is under budget
B‘(.4)
B is not under budget
B(.2)
A‘(.75)
A is not under budget
B is under budget
B‘(.8)
B is not under budget
Figure 4.6 Identifying the Relevant End
Nodes On The Tree
STAGE 1
PROJECT A
PERFORMANCE
STAGE 2
PROJECT B
PERFORMANCE
B(.6)
A (.25)
A is under budget
(1)
B is under budget
B‘(.4)
(2) 
B is not under budget
B(.2)
A‘(.75)
A is not under budget
B is under budget
B‘(.8)
B is not under budget
(3) 
(4)
Figure 4.7 Calculating End Node Probabilities
STAGE 1
PROJECT A
PERFORMANCE
STAGE 2
PROJECT B
PERFORMANCE
B(.6)
A (.25)
A is under budget
B is under budget
B‘(.4)
(1)
(2) .10
B is not under budget
B(.2)
A‘(.75)
A is not under budget
B is under budget
B‘(.8)
B is not under budget
(3)
(4)
.15
Figure 4.8
Probability Tree for the
Spare Parts Example
SOURCE
CONDITION
B (.04)
A (.7)
1
Adams is the supplier
Unit is defective
B' (.96)
Unit is OK
B (.07)
A
2
(.3)
Alder is the supplier
Unit is defective
B' (.93)
Unit is OK
Figure 4.9 Using the Tree to Calculate
End-Node Probabilities
SOURCE
CONDITION
B (.04)
A 1 (.7)
Adams is the supplier
Unit is defective

.028

.021
B' (.96)
Unit is OK
B (.07)
A 2 (.3)
Unit is defective
Alder is the supplier
B' (.93)
Unit is OK
Bayes’ Theorem
(Two Events)
P( A1 ) P( B / A1 )
P(A1 l B) =
P( A1 ) P( B / A1 )  P( A2 ) P( B / A2 )
(4.9)
General Form of Bayes’ Theorem
P( Ai ) P( B / Ai )
P(Ai l B) =
P( A1 ) P( B / A1 )  P( A2 ) P( B / A2 )  ...  P( Ak ) P( B / Ak )
Cross-tabs Table
Under
Grad
Very Important
Important
Not
Important
80
60
40
180
100
50
70
220
180
110
110
400
Grad
Joint Probability Table
Very
Important
Important
Not
Important
Under
Grad
.20
.15
.10
.45
Grad
.25
.125
.175
.55
.45
.275
.275
1.00
Counting Total Outcomes
(4.10)
in a Multi-Stage Experiment
Total Outcomes = m1 x m2 x m3 x…x mk
where mi = number of outcomes possible in each stage
k = number of stages
Combinations
(4.11)
n!
nCx =
(n  x)!x!
where
= number of combinations (subgroups) of n
objects selected x at a time
n = size of the larger group
x = size of the smaller subgroups
nCx
Figure 4.10
Your Car and Your Friends
Friends
2
3
A B
E
1
4
C
D
Car
Five friends are waiting for a ride in your car, but only four seats are available.
How many different arrangements of friends in the car are possible?
Permutations
n!
nPx =
(n  x )!
(4.12)