2. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Chapter 15 Probability Rules!

3. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 3 The General Addition Rule When two events A and B are disjoint, we can use the addition rule for disjoint events from Chapter 14: P(A  B) = P(A) + P(B) Why does this not work when they are not disjoint (a.k.a. joint)? I like that. Thus, we need the General Addition Rule. Let’s look at a picture…

4. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 4 The General Addition Rule (cont.) General Addition Rule: For any two events A and B, P(A  B) = P(A) + P(B) – P(A  B) The following Venn diagram shows a situation in which we would use the general addition rule: EX: Drawing a(n)… (a) ace or a spade (b) even # or a red (c) club or a diamond

5. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 5 It Depends… Back in Chapter 3, we looked at contingency tables and talked about conditional distributions. BLAST FROM THE PAST!!!!!!!!!!!! When we want the probability of an event from a conditional distribution, we write P(B|A) and pronounce it “the probability of B given A.” A probability that takes into account a given condition is called a conditional probability. This is used when two events are dependent.

6. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 6 It Depends… (cont.) To find the probability of the event B given the event A, we restrict our attention to the outcomes in A. We then find the fraction of those outcomes B that also occurred. Note: P(A) cannot equal 0, since we know that A has occurred.

7. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Direction Matters P(heart | red) P(red | heart) Slide 15 - 7

8. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 8 The General Multiplication Rule When two events A and B are independent, we can use the multiplication rule for independent events from Chapter 14: P(A  B) = P(A) x P(B) However, when our events are not independent, this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule.

9. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 9 The General Multiplication Rule (cont.) We encountered the general multiplication rule in the form of conditional probability. Rearranging the equation in the definition for conditional probability, we get the General Multiplication Rule: For any two events A and B, P(A  B) = P(A)  P(B|A) or P(A  B) = P(B)  P(A|B) Where do these come from?

10. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 10 Independence Independence of two events means that the outcome of one event does not influence the probability of the other. With our new notation for conditional probabilities, we can now formalize this definition: Events A and B are independent whenever P(B|A) = P(B). (Equivalently, events A and B are independent whenever P(A|B) = P(A).) Another BLAST FROM THE PAST!!!!!!!!

11. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 11 Independent ≠ Disjoint Disjoint events cannot be independent! Well, why not? Since we know that disjoint events have no outcomes in common, knowing that one occurred means the other didn’t. Thus, the probability of the second occurring changed based on our knowledge that the first occurred. It follows, then, that the two events are not independent. A common error is to treat disjoint events as if they were independent, and apply the Multiplication Rule for independent events—don’t make that mistake.

12. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 12

13. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 13 Depending on Independence It’s much easier to think about independent events than to deal with conditional probabilities. It seems that most people’s natural intuition for probabilities breaks down when it comes to conditional probabilities. Don’t fall into this trap: whenever you see probabilities multiplied together, stop and ask whether you think they are really independent.

14. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 14 Drawing Without Replacement Sampling without replacement means that once one individual is drawn it doesn’t go back into the pool. Drawing without replacement is just another instance of working with conditional probabilities.

15. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 15 Tree Diagrams A tree diagram helps us think through conditional probabilities by showing sequences of events as paths that look like branches of a tree. Making a tree diagram for situations with conditional probabilities is consistent with our “make a picture” mantra.

16. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Example with Tree Diagram According to a study, 44% of college students engage in binge drinking, 37% drink moderately, and 19% abstain completely. Another study finds that 17% of binge drinkers aged 21 to 34 have caused an alcohol-related car accident while 9% of moderate drinkers have caused such an accident. Slide 15 - 16

17. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 17 Tree Diagrams (cont.) We can add the final probabilities to find probabilities of compound events. Find the probability that someone chosen at random is a binge drinker and has caused an alcohol related accident. Moderate and accident? Abstain and accident?

18. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Reversing the Conditioning Reversing the conditioning of two events is rarely intuitive. Suppose we want to know P(A|B), and we know only P(A), P(B), and P(B|A). How would you accomplish this? Let P(A) = 40%, P(B) = 80%, and P(B|A) = 60%. What is P(A|B)? Slide 15 - 18

19. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 19 Reversing the Conditioning We know P(A  B), since P(A  B) = P(A) x P(B|A) From this information, we can find P(A|B): Apply this to the drinking example. From the diagram, I can easily find the probability that someone causes an alcohol-related accident given that they are a binge drinker. What is the probability that someone is a binge drinker given that they have caused an alcohol-related accident?

20. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Example A study done in Maryland found that in 77% of all accidents, the driver was wearing a seatbelt. Accident reports indicated that 92% of belt-wearers escaped with no serious injury, but only 63% of non-belted drivers were so fortunate. What’s the probability that a driver who was seriously injured wasn’t wearing a seatbelt? Slide 15 - 20

21. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 21 Bayes’s Rule When we reverse the probability from the conditional probability that you’re originally given, you are actually using Bayes’s Rule. WE WILL NEVER, EVER USE THIS.

22. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 22 What Can Go Wrong? Don’t use a simple probability rule where a general rule is appropriate: Don’t assume that two events are independent or disjoint without checking that they are. Don’t find probabilities for samples drawn without replacement as if they had been drawn with replacement. Don’t reverse conditioning naively. Don’t confuse “disjoint” with “independent.”

23. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 23 What have we learned? The probability rules from Chapter 14 only work in special cases—when events are disjoint or independent. We now know the General Addition Rule and General Multiplication Rule. We also know about conditional probabilities and that reversing the conditioning can give surprising results.

24. Copyright © 2010, 2007, 2004 Pearson Education, Inc. Slide 15 - 24 What have we learned? (cont.) Venn diagrams, tables, and tree diagrams help organize our thinking about probabilities. We now know more about independence—a sound understanding of independence will be important throughout the rest of this course.