Statistics for business decision making and analysis robert stine and foster chapter 08

8.1 From Tables to ProbabilityConverting Counts to Probabilities behaves like a random choice from the 17,619 cases in the contingency table fractions probabilities... 8.1 From Tables to

Conditional Probability

Chapter 8

8.1 From Tables to Probability

How does education affect income?

 Percentages computed within rows or

columns of a contingency table correspond

to conditional probabilities

 Conditional probabilities allow us to answer questions like how education affects income

8.1 From Tables to Probability

Contingency Table (Counts) for Amazon.com

8.1 From Tables to Probability

Converting Counts to Probabilities

behaves like a random choice from the

17,619 cases in the contingency table

fractions (probabilities)

8.1 From Tables to Probability

Probabilities for Amazon.com

8.1 From Tables to Probability

Joint Probability

 Displayed in cells of a contingency table

 Represent the probability of an intersection of two or more events

 For Amazon.com there are six joint

probabilities; e.g., P(Yes and MSN) = 0.016

8.1 From Tables to Probability

Marginal Probability

 Displayed in the margins of a contingency table

 Is the probability of observing an outcome with a single attribute, regardless of its other attributes

 For Amazon.com there are four marginal

probabilities, e.g., P(MSN) = 0.396 + 0.016 = 0.412

8.1 From Tables to Probability

Conditional Probability

given B, is P(A and B) / P(B)

restrict the sample space to a particular row

or column

8.1 From Tables to Probability

Conditional Probability

 Of interest to Amazon.com is the question

“which host will deliver the best visitors, those who are more likely to make a purchase?”

 Find conditional probabilities to answer

questions like “among visitors from MSN,

what is the chance a purchase is made?”

8.1 From Tables to Probability

Conditional Probability –

Restrict Sample Space to MSN

8.1 From Tables to Probability

Conditional Probability –

Compute Percentages within MSN Column

8.1 From Tables to Probability

Conditional Probabilities Show Purchases are more likely from MSN and Yahoo

P(Yes І MSN) = P(Yes and MSN) / P(MSN)

= 0.016 / 0.412 = 0.039

P(Yes І RecipeSource) ≈ 0.000

P(Yes І Yahoo) = 0.038

8.2 Dependent Events

Definition

Events that are not independent, indicated by

P(A and B) ≠ P(A) P(B) or

P(A) ≠ P(A І B)

×

8.2 Dependent Events

The Multiplication Rule

 Events in business tend to be dependent

(e.g., probability of purchasing a service given an ad for the

service is seen)

 Order matters:

Generally, P(A І B) ≠ P(B І A)

8.2 Dependent Events

The Multiplication Rule

The joint probability of two events A and B is the product of the marginal

probability of one times the conditional probability of the other

P(A and B) = P(A) x P(B І A)

P(A and B) = P(B) x P(A І B)

8.2 Dependent Events

The Multiplication Rule

 Disjoint events are never independent

 If A and B are disjoint, then

P(A І B) = P(A and B) / P(B)

= 0 / P(B) = 0

≠ P(A)

8.3 Organizing Probabilities

Probability Trees (Tree Diagrams)

 Graphical depiction of conditional probabilities (helpful for large

problems)

 Shows sequence of events as paths that suggest branches of a tree

8.3 Organizing Probabilities

Success of Advertising on TV

Programs Viewed on Sunday Evening

8.3 Organizing Probabilities

Success of Advertising on TV

Whether or Not Viewer Sees Ad

8.3 Organizing Probabilities

Use Tree Diagram to Find Probabilities

P(Watch game and See Ads) = 0.50 0.50

8.3 Organizing Probabilities

Derive Probability Table from Tree Diagram

Fill in Marginal Probabilities

8.3 Organizing Probabilities

Derive Probability Table from Tree Diagram

Fill in First Row of Joint Probabilities

8.3 Organizing Probabilities

Completed Probability Table

8.3 Order in Conditional Probabilities

If a viewer sees the ads, what is the chance she is watching Desperate

Housewives?

Find P(Desperate Housewives І See Ads)

= P(Desperate Housewives and See Ads)

P(See Ads)

= 0.07 / 0.455 = 0.154

4M Example 8.1:

DIAGNOSTIC TESTING

Motivation

If a mammogram indicates that a 55 year

old woman tests positive for breast cancer, what is the probability that she in fact has

breast cancer?

4M Example 8.1:

DIAGNOSTIC TESTING

Method

Past data indicates the following probabilities:

P(Test negative І No cancer) = 0.925

P(Test positive І Cancer) = 0.85

P(Cancer) = 0.003

4M Example 8.1:

DIAGNOSTIC TESTING

Mechanics – Fill in the Probability Table

4M Example 8.1:

DIAGNOSTIC TESTING

Mechanics – Fill in the Probability Table

Use Multiplication Rule to obtain joint

probabilities

For example, P (Cancer and Test positive)

= P (Cancer) P(Test positive І Cancer)

= 0.0030 0.85 = 0.00255

×

×

4M Example 8.1:

DIAGNOSTIC TESTING

Mechanics – Completed Probability Table

4M Example 8.2:

FILTERING JUNK MAIL

Method

Past data indicates the following probabilities:

P(Nigerian general І Junk mail) = 0.20

P(Nigerian general І Not Junk mail) = 0.001

P(Junk mail) = 0.50

4M Example 8.2:

FILTERING JUNK MAIL

Mechanics – Fill in the Probability Table

4M Example 8.2:

FILTERING JUNK MAIL

Mechanics –

Use Table to find Conditional Probability

P (Junk mail І Nigerian general)

= 0.1 / 0.1005

= 0.995

4M Example 8.2:

FILTERING JUNK MAIL

Message

Email messages to this employee with the

phrase “Nigerian general” have a high

probability (more than 99%) of being spam

Best Practices

the Multiplication Rule

Best Practices (Continued)

the same thing as “independent.”