716 CHAPTER 11 Probability In many of the examples, we used the words given that. Other words may be used instead. For example, in Example 3(b), the question could have been worded “was satisfied with their surgery if the person had knee surgery.” Recall from Section 11.5 that two events, A and B, are independent events if the occurrence of either event in no way affects the probability of the occurrence of the other event. Consider two independent events, A and B, and P A B ( ). Since the occurrence of event B in no way affects the probability of the occurrence of event A P A B P A , ( ) ( ). = Likewise, consider P B A ( ). Since the occurrence of event A in no way affects the probability of the occurrence of event B P B A P B , ( ). ( ). = This leads us to an alternative definition of independent events. surgery, n E( ), 1 is 105. The number of people who had hip surgery and were dissatisfied with the results of the surgery, n E E ( and ), 1 2 is 15. Thus, = = = P E E n E E n E ( ) ( and ) ( ) 15 105 1 7 2 1 1 2 1 d) We are given that the person was dissatisfied with the results of the surgery. Thus, we have a conditional probability problem. Let E1 be the given information “the person was dissatisfied with the results of the surgery.” Let E2 be “the person had heart surgery.” We are asked to determine P E E ( ). 2 1 The number of people who were dissatisfied with the results of the surgery, n E( ), 1 is 40. The number of people who were dissatisfied with the results of the surgery and had heart surgery, n E E ( and ), 1 2 is 5. Thus, = = = P E E n E E n E ( ) ( and ) ( ) 5 40 1 8 2 1 1 2 1 7 Now try Exercise 43 Learning Catalytics Keyword: Angel-SOM-11.6 (See Preface for additional details.) Definition: Independent Events Event A and Event B are independent events if and only if P A B P A ( ) ( ) = and P B A P B ( ) ( ). = We will discuss independent events more throughout the rest of this chapter. Exercises 75 and 77 also involve independent events. Instructor Resources for Section 11.6 in MyLab Math • Objective-Level Videos 11.6 • Interactive Concept Video: Conditional Probability • StatCrunch: Games: Let's Make a Deal • PowerPoint Lecture Slides 11.6 • MyLab Exercises and Assignments 11.6 Exercises Warm Up Exercises In Exercises 1– 4, fill in the blank with an appropriate word, phrase, or symbol(s). 1. The probability of event E2 occurring, given that event E1 has happened, is called a(n) ________ probability. Conditional 2. The notation for the probability of E ,2 given E ,1 is ________. P E E ( ) 2 1 3. If n E E ( and ) 3 1 2 = and n E( ) 7, 1 = then P E E ( ) 2 1 = ________. 3 7 4. If n E E ( and ) 4 1 2 = and n E( ) 16, 1 = then P E E ( ) 2 1 = ________. 1 4 Practice the Skills Drawing a Card In Exercises 5–8, one card is drawn from a standard 52-card deck. Determine the probability that the card is a 5. a) a club. 1 4 b) a club, given that the card is a black card. 1 2 6. a) a queen. 1 13 b) a queen, given that the card is a face card. 1 3 7. a) a jack or a king. 2 13 b) a jack or a king, given that the card is a face card. 2 3 SECTION 11.6 Applets Games Let’s Make a Deal StatCrunch
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