copyright © 2014, 2011 pearson education, inc. 1 chapter 8 conditional probability

Copyright © 2014, 2011 Pearson Education, Inc. 1

Chapter 8Conditional Probability


8.1 From Tables to Probabilities

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



Contingency Table (Counts) for Amazon.com



Converting Counts to Probabilities

Assume the next visitor to Amazon.com behaves like a random choice from the 28,975 cases in the contingency table

Divide each count by 28,975 to get fractions (probabilities)



Probabilities for Amazon.com



Joint Probability

Displayed in cells of a contingency table

Represent the probability of an intersection of two or more events (combination of attributes)

For Amazon.com there are six joint probabilities; e.g., P(Yes and Comcast) = 0.001



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 five marginal probabilities, e.g., P(Comcast) = 0.009 + 0.001 = 0.010



Conditional Probability

P(A І B), the conditional probability of A given B, is P(A and B) / P(B)

To obtain a conditional probability, we restrict the sample space to a particular row or column



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 Comcast, what is the chance a purchase is made?”



Conditional Probability – Restrict Sample Space to Comcast



Conditional Probability – Compute Percentages in Comcast Column



Conditional Probabilities –Purchases more likely from Comcast

P(Yes І Comcast) = P(Yes and Comcast)P(Comcast)

= 0.001 / 0.010 = 0.100

P(Yes І Google) = 0.033P(Yes І Nextag) = 0.042


8.2 Dependent Events

Definition

Events that are not independent; for dependent events P(A and B) ≠ P(A)×P(B)

or P(A) ≠ P(A І B)



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)




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) × P(B І A)P(A and B) = P(B) × P(A І B)




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



Success of Advertising on TVPrograms Viewed on Sunday Evening



Success of Advertising on TVWhether or Not Viewer Sees Ad



Use Tree Diagram to Find Probabilities

P(Watch game and See Ads) = 0.50 0.50= 0.25

P(See Ads) = 0.15 0.90 + 0.35 0.20 + 0.50 0.50

= 0.455



Derive Probability Table from Tree DiagramFill in Marginal Probabilities



Derive Probability Table from Tree DiagramFill in First Row of Joint Probabilities



Completed Probability Table


8.4 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?



Method

Past data indicates the following probabilities:

P(Test negative І No cancer) = 0.925P(Test positive І Cancer) = 0.85P(Cancer) = 0.005



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)= 0.005 0.85 = 0.00425



Mechanics – Completed Probability Table



Message

The chance that a woman who tests positive actually has cancer is small, a bit more than 5%.



Bayes’ Rule: Reversing a Conditional Probability Algebraically

P(A І B) = _____P(B І A) P(A)______P(B І A) P(A) + P(B І Ac) P(Ac)


4M Example 8.2: FILTERING JUNK MAIL

Motivation

Is there a way to help workers filter out junk mail from important email messages?



Method

Past data indicates the following probabilities:

P(Nigerian general І Junk mail) = 0.20P(Nigerian general І Not Junk mail) = 0.001P(Junk mail) = 0.50



Mechanics – Use Table to find Conditional Probability

P (Junk mail І Nigerian general) = 0.1 / 0.1005= 0.995



Message

Email messages to this employee with the phrase “Nigerian general” have a high probability (more than 99%) of being spam.


Best Practices

Think conditionally.

Presume events are dependent and use the Multiplication Rule.

Use tables to organize probabilities.


Best Practices (Continued)

Use probability trees for sequences of conditional probabilities.

Check that you have included all of the events.

Use Bayes’ Rule to reverse the order of conditioning.


Pitfalls

Do not confuse P(A І B) for P(B І A).

Don’t think that “mutually exclusive” means the same thing as “independent.”

Do not confuse counts with probabilities.

copyright © 2014, 2011 pearson education, inc. 1 chapter 8 conditional probability

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