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

2. Copyright © 2014, 2011 Pearson Education, Inc. 2 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

3. Copyright © 2014, 2011 Pearson Education, Inc. 3 8.1 From Tables to Probabilities Contingency Table (Counts) for Amazon.com

4. Copyright © 2014, 2011 Pearson Education, Inc. 4 8.1 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)

5. Copyright © 2014, 2011 Pearson Education, Inc. 5 8.1 From Tables to Probabilities Probabilities for Amazon.com

6. Copyright © 2014, 2011 Pearson Education, Inc. 6 8.1 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

7. Copyright © 2014, 2011 Pearson Education, Inc. 7 8.1 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) = 0.009 + 0.001 = 0.010

8. Copyright © 2014, 2011 Pearson Education, Inc. 8 8.1 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

9. Copyright © 2014, 2011 Pearson Education, Inc. 9 8.1 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?”

10. Copyright © 2014, 2011 Pearson Education, Inc. 10 8.1 From Tables to Probabilities Conditional Probability – Restrict Sample Space to Comcast

11. Copyright © 2014, 2011 Pearson Education, Inc. 11 8.1 From Tables to Probabilities Conditional Probability – Compute Percentages in Comcast Column

12. Copyright © 2014, 2011 Pearson Education, Inc. 12 8.1 From Tables to Probabilities 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) = 0.033 P(Yes І Nextag) = 0.042

13. Copyright © 2014, 2011 Pearson Education, Inc. 13 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)

14. Copyright © 2014, 2011 Pearson Education, Inc. 14 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)

15. Copyright © 2014, 2011 Pearson Education, Inc. 15 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) × P(B І A) P(A and B) = P(B) × P(A І B)

16. Copyright © 2014, 2011 Pearson Education, Inc. 16 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)

17. Copyright © 2014, 2011 Pearson Education, Inc. 17 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

18. Copyright © 2014, 2011 Pearson Education, Inc. 18 8.3 Organizing Probabilities Success of Advertising on TV Programs Viewed on Sunday Evening

19. Copyright © 2014, 2011 Pearson Education, Inc. 19 8.3 Organizing Probabilities Success of Advertising on TV Whether or Not Viewer Sees Ad

20. Copyright © 2014, 2011 Pearson Education, Inc. 20 8.3 Organizing Probabilities 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

21. Copyright © 2014, 2011 Pearson Education, Inc. 21 8.3 Organizing Probabilities Derive Probability Table from Tree Diagram Fill in Marginal Probabilities

22. Copyright © 2014, 2011 Pearson Education, Inc. 22 8.3 Organizing Probabilities Derive Probability Table from Tree Diagram Fill in First Row of Joint Probabilities

23. Copyright © 2014, 2011 Pearson Education, Inc. 23 8.3 Organizing Probabilities Completed Probability Table

24. Copyright © 2014, 2011 Pearson Education, Inc. 24 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

25. 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?

26. 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) = 0.925 P(Test positive І Cancer) = 0.85 P(Cancer) = 0.005

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

28. 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) =0.005 0.85 = 0.00425

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

30. 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%.

31. Copyright © 2014, 2011 Pearson Education, Inc. 31 8.4 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 )

32. 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 email messages?

33. 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) = 0.001 P(Junk mail) = 0.50

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

35. 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 / 0.1005 = 0.995

36. Copyright © 2014, 2011 Pearson Education, Inc. 36 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.

37. 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.

38. 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.

39. 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.