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
P(A) =
where P(A) = probability of event A
F = number of outcomes “favorable” to event A
T = total number of outcomes possible in the experiment
T
F
Relative Frequency Approach
P(A) =
where N = total number of observations
or trials
n = number of times that event A
occurs
N
n
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.
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.
“Conditional Equals JOINT Over SIMPLE” Rule
)(
)(
AP
BAP P(B I A) = (4.7a)
P(A I B) = (4.7b))(
)(
BP
BAP
Figure 4.1 Venn Diagram for the Internet Shoppers Example
A(.8)
(Airline Ticket Purchase)
B(.6)
(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.
Sample Space (1.0)
Figure 4.2 Complementary Events
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).
Sample Space (1.0)
A
.8
A’ (.2)
(everything in the sample space outside A)
Figure 4.3 Mutually Exclusive Events
Mutually exclusive events appear as non-overlapping circles in a Venn diagram.
Sample Space (1.0)
BA
Figure 4.4 Probability Tree for the Project Example
B is not under budget
A
A'
A is under budget
B is under budget
B
B'
A is not under budget
B is under budget
B is not under budget
B
PROJECT A PERFORMANCE
PROJECT B PERFORMANCE
B'
STAGE 1 STAGE 2
Figure 4.5 Showing Probabilities on the Tree
B is not under budget
A (.25)
A‘(.75)
A is under budget
B is under budget
B(.6)
B‘(.4)
A is not under budget
B is under budget
B is not under budget
B(.2)
PROJECT A PERFORMANCE
PROJECT B PERFORMANCE
B‘(.8)
STAGE 1 STAGE 2
Figure 4.6 Identifying the Relevant End Nodes On The Tree
B is not under budget
A (.25)
A‘(.75)
A is under budget
B is under budget
B(.6)
B‘(.4)
A is not under budget
B is under budget
B is not under budget
B(.2)
PROJECT A PERFORMANCE
PROJECT B PERFORMANCE
B‘(.8)
STAGE 1 STAGE 2
(2)
(1)
(3)
(4)
Figure 4.7 Calculating End Node Probabilities
B is not under budget
A (.25)
A‘(.75)
A is under budget
B is under budget
B(.6)
B‘(.4)
A is not under budget
B is under budget
B is not under budget
B(.2)
PROJECT A PERFORMANCE
PROJECT B PERFORMANCE
B‘(.8)
STAGE 1 STAGE 2
(2)
(1)
(3)
(4)
.10
.15
Figure 4.8 Probability Tree for the Spare Parts Example
Unit is OK
A1
A2
Adams is the supplier
Unit is defective
B
B'
Alder is the supplier
Unit is defective
Unit is OK
B
B'
(.7)
(.3)
(.04)
(.96)
(.07)
(.93)
SOURCE CONDITION
Figure 4.9 Using the Tree to Calculate End-Node Probabilities
Unit is OK
A 1
A2
Adams is the supplier
Unit is defective
B
B'
Alder is the supplier
Unit is defective
Unit is OK
B
B'
(.7)
(.3)
(.04)
(.96)
(.07)
(.93)
SOURCE CONDITION
.028
.021
General Form of Bayes’ Theorem
)/()(...)/()()/()(
)/()(
2211 kk
ii
ABPAPABPAPABPAP
ABPAP
P(Ai l B) =
Cross-tabs Table
Very Important
Important NotImportant
Under Grad 80 60 40 180
Grad 100 50 70 220
180 110 110 400
Joint Probability Table
Very Important
Important NotImportant
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)
!)!(
!
xxn
n
nCx =
where nCx = number of combinations (subgroups) of n objects selected x at a time n = size of the larger group x = size of the smaller subgroups
Figure 4.10 Your Car and Your Friends
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?
2
3
4
A B E C D
1
Friends
Car