2. Copyright © 2009 Pearson Education, Inc. Chapter 15 Probability Rules!

3. Copyright © 2009 Pearson Education, Inc. Slide 1- 3 Objectives: The student will be able to: Apply the General Addition and General Multiplication rules. Compute conditional probabilities and use the rule to test for independence. Know how to make and use a tree diagram to understand conditional probabilities and reverse conditioning.

4. Copyright © 2009 Pearson Education, Inc. Slide 1- 4 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 or B) = P(A) + P(B) However, when our events are not disjoint, this earlier addition rule will double count the probability of both A and B occurring. Thus, we need the General Addition Rule.

5. Copyright © 2009 Pearson Education, Inc. Slide 1- 5 The General Addition Rule (cont.) General Addition Rule: For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B) The following Venn diagram shows a situation in which we would use the general addition rule:

6. Copyright © 2009 Pearson Education, Inc. Slide 1- 6 The General Addition Rule (cont.) General Addition Rule: For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B) Example: you roll a fair die What is the probability of rolling an even number or a number greater than 3? Example: 50% of the students in statistics have taken another college math class, 60% have taken an English class, and 30% have taken both. What is the probability that a student chosen at random has taken a math or an English class? Hint: use a Venn Diagram to help!

7. Copyright © 2009 Pearson Education, Inc. Practice College dorm rooms on a large college campus had the following: 38% refrigerators, 52% tvs, and 21% had both. What is the probability that a randomly selected dorm room had: A tv but no refrigerator? A TV or a refrigerator but not both? Neither a tv nor a refrigerator? Employment data at a large company revealed that 72% of workers are married, 44% are college graduates, and half of the college graduates are married. What’s the probability that a randomly selected worker Is neither married nor a college graduate? Is married but not a college graduate? Is married or a college graduate? Slide 1- 7

8. Copyright © 2009 Pearson Education, Inc. Slide 1- 8 Using Contingency Tables to determine Probability Back in Chapter 3, we looked at contingency tables and talked about conditional distributions, it is a short step from relative frequencies to probabilities. What is the P(Boy)? P(Popular)? P(Boy or Girl)? P(Grades or Sports)? P(Boy and Popular)? (Why don’t we use multiplication rule here?) ***GradesPopularSportsTotal Boy1175060227 Girl1309130251 Total24714190478 *** Data is from student reports on what was most important to them, Grades, being Popular, or doing well in Sports

9. Copyright © 2009 Pearson Education, Inc. Slide 1- 9 It Depends… Conditional Probability 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. To find the probability of the event B given the event A, we restrict our attention to the outcomes in A. We then find in what fraction of those outcomes B also occurred. Note: P(A) cannot equal 0, since we know that A has occurred.

10. Copyright © 2009 Pearson Education, Inc. Slide 1- 10 It Depends… P(girl) = (Total girls)/Total = 251/478 = 0.525 P(girl and popular) = (# girls and popular)/Total = 91/478 = 0.190 P(sports) = (Total Sports)/Total = 90/478 = 0.188 P(sports | girl) = (girls and sports)/girls = 30/251 = 0.120 P(sports | boy) = (boys and sports)/boys = 60/227 = 0.264 GradesPopularSportsTotal Boy1175060227 Girl1309130251 Total24714190478

11. Copyright © 2009 Pearson Education, Inc. The probabilities that an adult man has high blood pressure and/or high cholesterol are shown in the table: What is the probability that a man has both conditions? What is the probability that he has high blood pressure? What is the probability that a man with high blood pressure has high cholesterol What’s the probability that a man has high blood pressure if it’s known that he has high cholesterol? Slide 1- 11 High BP OK BP High Chol.0.110.21 OK. Chol.0.160.52

12. Copyright © 2009 Pearson Education, Inc. Slide 1- 12 It Depends… Practice: You draw a card at random from a standard deck of 52 cards. Find each conditional probability The card is a heart, given that it is red The card is red, given that it is a heart The card is an ace, given that it is red The card is a queen, given that it is a face card

13. Copyright © 2009 Pearson Education, Inc. Slide 1- 13 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).) Example: In the previous example having a goal of sports and being a Girl are not independent because P(sports) does not equal P(sports|Girl)

14. Copyright © 2009 Pearson Education, Inc. If you draw a card at random from a well-shuffled deck is the probability of getting an ace independent of the suit? E.g. Does P(ace) = P(ace | suit is hearts)? Approximately 58.2% of US adults have both a landline an a cell phone. 2.8% have only cell phone, 1.6% have no phone service at all What proportion of US households have a landline? Are having a cell phone and having a landline independent? Does P(cell phone)=P(cell phone | have a landline)? Or P(landline)=P(landline | cell phone)? Slide 1- 14

15. Copyright © 2009 Pearson Education, Inc. Slide 1- 15 The General Multiplication Rule When two events A and B are independent, we can use the multiplication rule (Light) for independent events from Chapter 14: P(A and 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.

16. Copyright © 2009 Pearson Education, Inc. Slide 1- 16 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 and B) = P(A) x P(B|A) or P(A and B) = P(B) x P(A|B)

17. Copyright © 2009 Pearson Education, Inc. Slide 1- 17 Apply the General Multiplication Rule For any two events A and B, P(A and B) = P(A) x P(B|A) or P(A and B) = P(B) x P(A|B) 70% of kids who visit a doctor have a fever and 30% of kids with a fever have sore throats. What is the probability that a kid who goes to the doctor has a fever and a sore throat? If there are 6 balls in a bag, 3 yellow, 2 green, and 1 red. If you pick one ball and don’t replace it, what is the probability that you draw a yellow ball and then a green ball?

18. Copyright © 2009 Pearson Education, Inc. Slide 1- 18 Drawing Without Replacement Sampling without replacement means that once one object is drawn it doesn’t go back into the pool. We often sample without replacement, which doesn’t matter too much when we are dealing with a large population. However, when drawing from a small population, we need to take note and adjust probabilities accordingly. Drawing without replacement is just another instance of working with conditional probabilities. Example: text #15: You are dealt a hand of 3 cards one at a time. What is the Probability The first heart you get is the 3 rd card dealt? Your cards all are red You get no spades You have at least one ace

19. Copyright © 2009 Pearson Education, Inc. Slide 1- 19 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.

20. Copyright © 2009 Pearson Education, Inc. Slide 1- 20 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.

21. Copyright © 2009 Pearson Education, Inc. Slide 1- 21 Tree Diagrams (cont.) Figure 15.5 is a nice example of a tree diagram and shows how we multiply the probabilities of the branches together. All the final outcomes are disjoint and must add up to one. We can add the probabilities to find probabilities of compound events.

22. Copyright © 2009 Pearson Education, Inc. Slide 1- 22 Tree Diagrams Practice – text #42 using a tree diagram An airline offers discounted “advance-purchase” fares to customers who buy tickets more than 30 days before travel and charges “regular” fares for tickets purchased during those last 30 days. The company has noticed that 60%of its customers buy advance purchase tickets. The “no-show” rate among people who purchase regular tickets is 30% but only 5% of customers with advance tickets are no-shows. What percent of all ticket holders are no-shows? What’s the probability that an customer who didn’t show had an advance-purchase ticket? Is being a no-show independent of the type of ticket a passenger holds?

23. Copyright © 2009 Pearson Education, Inc. Slide 1- 23 Reversing the Conditioning Motivating example: A diagnostic test is 98% accurate meaning that it has a false positive rate of 2%. T false negative rate is.1%. If the illness is rare, affection only 3 in 100,000, what is the probability that I have the illness given that I tested positive? I know P(+ test | not sick) and P(- test | sick) but I want to know the P(sick | + test). Lets make a tree diagram. Recall that P(sick | + test) = P(sick and + test) / P( + test)

24. Copyright © 2009 Pearson Education, Inc. Slide 1- 24 Reversing the Conditioning Reversing the conditioning of two events is rarely intuitive. Suppose we want to know P(A|B), but we know only P(A), P(B), and P(B|A). We also know P(A and B), since P(A and B) = P(A) x P(B|A) From this information, we can find P(A|B):

25. Copyright © 2009 Pearson Education, Inc. Slide 1- 25 Reversing the Conditioning (cont.) When we reverse the probability from the conditional probability that you’re originally given, you are actually using Bayes’s Rule. (don’t memorize this, use tree diagram instead) Practice: #46, 43

26. Copyright © 2009 Pearson Education, Inc. Slide 1- 26 Reversing the Conditioning (cont.) Practice: #46 Suppose a polygraph can detect 65% of lies, but incorrectly identifies 15% of true statements as lies A certain company believes that 95%of its job applicants are trustworthy. The company gives everyone a polygraph. What is the probability that a job applicant rejected due to failing the polygraph was actually trustworthy? Want: P(trustworthy | failed polygraph)

27. Copyright © 2009 Pearson Education, Inc. Slide 1- 27 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.”

28. Copyright © 2009 Pearson Education, Inc. Slide 1- 28 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.

29. Copyright © 2009 Pearson Education, Inc. Slide 1- 29 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.