1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 15 Probability Rules!

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 2 Recall That… For any random phenomenon, each trial generates an _______________. An ________ is any set or collection of outcomes. The collection of all possible outcomes is called the ___________________, denoted ___.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 3 Events When outcomes are ______________________, probabilities for events are easy to find just by counting. When the k possible outcomes are equally likely, each has a probability of ___. For any event A that is made up of equally likely outcomes,

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 4 The First Three Rules for Working with Probability Rules

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 5 Picturing Probabilities The most common kind of picture to make is called a _______ diagram. We will see _______ diagrams in practice shortly…

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 6 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) = However, when our events are not disjoint, this earlier addition rule will ____________________ the probability of both A and B occurring. Thus, we need the General Addition Rule.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 7 The General Addition Rule (cont.) General Addition Rule: For any two events A and B, P(A or B) = The following Venn diagram shows a situation in which we would use the general addition rule:

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 8 Just Checking Back in chapter 1 we suggested that you sample some pages of this book at random to see whether they had a graph or other data display. We actually did just that. We drew a representative sample and found the following: 48% of pages had some kind of data display 27% of pages had an equation, and 7% of pages had both a) Display these results in a Venn diagram. b) What is the probability that a randomly selected sample page had a data display but no equation? c) What is the probability that a randomly selected sample page had neither a data display nor an equation?

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 9 Assignment p. 362 #1, 4, 5, 8-12, 14-16

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 15 Probability Rules! (2)

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 11 It Depends… Back in Chapter 3, we looked at contingency tables and talked about ____________________. When we want the probability of an event from a conditional distribution, we write ______ and pronounce it “the probability of B given A.” A probability that takes into account a given condition is called a ______________________.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 12 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.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 13 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 and B) = ____________ However, when our events are ______________, this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 14 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) = ________________ or P(A and B) = ________________

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 15 Independence Independence of two events means that the ____________ of one event does not influence the _______________ of the other. With our new notation for conditional probabilities, we can now formalize this definition: Events A and B are independent whenever ________________ (Equivalently, events A and B are independent whenever ________________

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 16 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 ______ _____________________ Thus, the probability of the second occurring ________ based on our knowledge that the first occurred. It follows, then, that the two events are ________ _____________________ A common error is to treat disjoint events as if they were independent, and apply the _______________________ for independent events—don’t make that mistake.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 17 Depending on Independence It’s much easier to think about independent events than to deal with ___________________. It seems that most people’s natural intuition for probabilities breaks down when it comes to _________________________. Don’t fall into this trap: whenever you see probabilities multiplied together, stop and ask whether you think they are really ____________.

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 18 Just Checking Remember our sample of pages in this book from this chapter’s first Just Checking…? We found that 48% of pages had some kind of data display 27% of pages had an equation, and 7% of pages had both a) Make a contingency table for the variables display and equation. b) What is the probability that a randomly selected sample page with an equation also had a data display? c) Are having an equation and having a data display disjoint events? d) Are having an equation and having a data display independent events?

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 19 Drawing Without Replacement Sampling without replacement means that once one individual is drawn it _____________________________. We often sample without replacement, which doesn’t matter too much when we are dealing with _________ ___________________ However, when drawing from a small population, we need to ______________ and ___________________ accordingly. Drawing without replacement is just another instance of working with ____________________________

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 20 Tree Diagrams A tree diagram helps us think through conditional probabilities by showing sequences of events as paths that look like _______________________. Making a tree diagram for situations with ________________ probabilities is consistent with our “__________________” mantra.

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 21 Tree Diagrams (cont.) Figure 15.4 is a nice example of a tree diagram and shows how we multiply the probabilities of the branches together:

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 22 Assignment p.362 #2, 3, 6, 7, 13, 18, 19, 24, 25, 27, 32

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 15 Probability Rules! (3)

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 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) = From this information, we can find P(A|B):

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 25 Reversing the Conditioning (cont.) When we reverse the probability from the conditional probability that you’re originally given, you are actually using ____________________.

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 26 What Can Go Wrong? Don’t use a ___________ probability rule where a _______________ rule is appropriate: Don’t assume that two events are ________________ or ______________ without checking that they are. Don’t find probabilities for samples drawn ____________ replacement as if they had been drawn ___________ replacement. Don’t ___________ conditioning naively. Don’t confuse “_________” with “____________.”

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 27 What have we learned? The probability rules from Chapter 14 only work in special cases—when events are _________ or _______________. We now know the General _____________ Rule and General ______________ Rule. We also know about conditional probabilities and that ______________ the conditioning can give surprising results.

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 28 What have we learned? (cont.) Venn diagrams, tables, and tree diagrams help _____________ our thinking about probabilities. We now know more about _______________—a sound understanding of _________________ will be important throughout the rest of this course.

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Practice Given a standard deck of cards (52), find the probability of each. A)If one card is drawn, what is the probability that it is an Ace or Red? B)Two cards are drawn without replacement. What is the probability they are both Aces? C)I draw one card and look at it. I tell you it is red. What is the probability it is a heart? Also, what is the probability it is red, given it is a heart? D)Are a red card and a spade independent? Mutually exclusive? E) Are a red card and an ace independent? Mutually exclusive? F) Are a face card and a king independent? Mutually exclusive? Slide 15- 29

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 30 Assignment p. 364 #17, 20, 23, 31, 34, 35, 36, 38, 43, 44