Copyright © 2014, 2011 Pearson Education, Inc. 1 Chapter 8 Conditional Probability

Copyright © 2014, 2011 Pearson Education, Inc 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

Copyright © 2014, 2011 Pearson Education, Inc From Tables to Probabilities Contingency Table (Counts) for Amazon.com

Copyright © 2014, 2011 Pearson Education, Inc From Tables to Probabilities 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)

Copyright © 2014, 2011 Pearson Education, Inc From Tables to Probabilities Probabilities for Amazon.com

Copyright © 2014, 2011 Pearson Education, Inc From Tables to Probabilities 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

Copyright © 2014, 2011 Pearson Education, Inc From Tables to Probabilities 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) = = 0.010

Copyright © 2014, 2011 Pearson Education, Inc From Tables to Probabilities 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

Copyright © 2014, 2011 Pearson Education, Inc From Tables to Probabilities 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?”

Copyright © 2014, 2011 Pearson Education, Inc From Tables to Probabilities Conditional Probability – Restrict Sample Space to Comcast

Copyright © 2014, 2011 Pearson Education, Inc From Tables to Probabilities Conditional Probability – Compute Percentages in Comcast Column

Copyright © 2014, 2011 Pearson Education, Inc From Tables to Probabilities Conditional Probabilities – Purchases more likely from Comcast P(Yes І Comcast) = P(Yes and Comcast) P(Comcast) = / = P(Yes І Google) = P(Yes І Nextag) = 0.042

Copyright © 2014, 2011 Pearson Education, Inc 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)

Copyright © 2014, 2011 Pearson Education, Inc 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)

Copyright © 2014, 2011 Pearson Education, Inc 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) × P(B І A) P(A and B) = P(B) × P(A І B)

Copyright © 2014, 2011 Pearson Education, Inc 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)

Copyright © 2014, 2011 Pearson Education, Inc 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

Copyright © 2014, 2011 Pearson Education, Inc Organizing Probabilities Success of Advertising on TV Programs Viewed on Sunday Evening

Copyright © 2014, 2011 Pearson Education, Inc Organizing Probabilities Success of Advertising on TV Whether or Not Viewer Sees Ad

Copyright © 2014, 2011 Pearson Education, Inc Organizing Probabilities Use Tree Diagram to Find Probabilities P(Watch game and See Ads)= = 0.25 P(See Ads) = = 0.455

Copyright © 2014, 2011 Pearson Education, Inc Organizing Probabilities Derive Probability Table from Tree Diagram Fill in Marginal Probabilities

Copyright © 2014, 2011 Pearson Education, Inc Organizing Probabilities Derive Probability Table from Tree Diagram Fill in First Row of Joint Probabilities

Copyright © 2014, 2011 Pearson Education, Inc Organizing Probabilities Completed Probability Table

Copyright © 2014, 2011 Pearson Education, Inc 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 / = 0.154

Copyright © 2014, 2011 Pearson Education, Inc. 25 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?

Copyright © 2014, 2011 Pearson Education, Inc. 26 4M Example 8.1: DIAGNOSTIC TESTING Method Past data indicates the following probabilities: P(Test negative І No cancer) = P(Test positive І Cancer) = 0.85 P(Cancer) = 0.005

Copyright © 2014, 2011 Pearson Education, Inc. 27 4M Example 8.1: DIAGNOSTIC TESTING Mechanics – Fill in the Probability Table

Copyright © 2014, 2011 Pearson Education, Inc. 28 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) = =

Copyright © 2014, 2011 Pearson Education, Inc. 29 4M Example 8.1: DIAGNOSTIC TESTING Mechanics – Completed Probability Table

Copyright © 2014, 2011 Pearson Education, Inc. 30 4M Example 8.1: DIAGNOSTIC TESTING Message The chance that a woman who tests positive actually has cancer is small, a bit more than 5%.

Copyright © 2014, 2011 Pearson Education, Inc Organizing Probabilities 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 A c ) P(A c )

Copyright © 2014, 2011 Pearson Education, Inc. 32 4M Example 8.2: FILTERING JUNK MAIL Motivation Is there a way to help workers filter out junk mail from important messages?

Copyright © 2014, 2011 Pearson Education, Inc. 33 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) = P(Junk mail) = 0.50

Copyright © 2014, 2011 Pearson Education, Inc. 34 4M Example 8.2: FILTERING JUNK MAIL Mechanics – Fill in the Probability Table

Copyright © 2014, 2011 Pearson Education, Inc. 35 4M Example 8.2: FILTERING JUNK MAIL Mechanics – Use Table to find Conditional Probability P (Junk mail І Nigerian general) = 0.1 / = 0.995

Copyright © 2014, 2011 Pearson Education, Inc. 36 4M Example 8.2: FILTERING JUNK MAIL Message messages to this employee with the phrase “Nigerian general” have a high probability (more than 99%) of being spam.

Copyright © 2014, 2011 Pearson Education, Inc. 37 Best Practices  Think conditionally.  Presume events are dependent and use the Multiplication Rule.  Use tables to organize probabilities.

Copyright © 2014, 2011 Pearson Education, Inc. 38 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.

Copyright © 2014, 2011 Pearson Education, Inc. 39 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.