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

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.

Statistical Independence (4.1)

P(A l B) = P(A) and P (B l A) = P(B)

“given” “given”

General Multiplication Rule (4.2)

P(A B) = P(A)∙ P(BlA)

“and”

Multiplication Rule for (4.3)

Independent Events

P(A B) = P(A) ∙P(B)

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 (4.4)

P(A B) = 0

General Addition Rule (4.5)

P(A B) = P(A) + P(B) - P(A B)

“or”

Addition Rule for (4.6) Mutually Exclusive Events

P(A B) = P(A) + P(B)

“Conditional Equals JOINT Over SIMPLE” Rule

)(

)(

AP

BAP P(B I A) = (4.7a)

P(A I B) = (4.7b))(

)(

BP

BAP

Complementary Events Rule (4.8)

P(A′ ) = 1 – P(A)

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

Bayes’ Theorem (4.9) (Two Events)

)/()()/()(

)/()(

2211

11

ABPAPABPAP

ABPAP

P(A1 l B) =

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

Permutations (4.12)

n

n x

!

( )!nPx =