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Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. SECTION 4.2 THE ADDITION RULE AND THE RULE OF COMPLEMENTS Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objectives 1. 2. 3. Compute probabilities by using the General Addition Rule Compute probabilities by using the Addition Rule for Mutually Exclusive Events Compute probabilities by using the Rule of Complements Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objective 1 Compute probabilities by using the General Addition Rule Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Events in the Form A or B A compound event is an event that is formed by combining two or more events. One type of compound event is of the form A or B. The event A or B occurs whenever A occurs, B occurs, or A and B both occur. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Consider the following table which presents the result of a survey in which 1000 adults were asked whether they favored a law that would provide government support for higher education. Each person was also asked whether they voted in the last election. Favor Oppose Undecided Likely to vote 372 262 87 Not likely to vote 151 103 25 The event that a person is likely to vote or favors the law is an event in the form A or B. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The General Addition Rule To compute probabilities of the form P(A or B), we use the General Addition Rule. For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B) Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example Find the probability that a randomly selected person is likely to vote or favors the law. Favor Oppose Undecided Likely to vote 372 262 87 Not likely to vote 151 103 25 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Solution Favor Oppose Undecided Likely to vote 372 262 87 Not likely to vote 151 103 25 Using the General Addition Rule, we have P(Likely to vote OR Favors the law) = P(Likely to vote) + P(Favors the law) – P(Likely to vote AND Favors the law) Now, there are 372 + 262 + 87 = 721 people who are likely to vote so P(Likely to vote) = 721/1000 = 0.721. There are 372 + 151 = 523 people who favor the law so P(Favor the law) = 523/1000 = 0.523. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Solution Favor Oppose Undecided Likely to vote 372 262 87 Not likely to vote 151 103 25 The number of people who are both likely to vote and who favor the law is 372. Therefore, P(Likely to vote AND Favors the law) = 372/1000 = 0.372. By the General Addition Rule, P(Likely to vote OR Favors the law) = P(Likely to vote) + P(Favors the law) – P(Likely to vote AND Favors the law) = 0.721 + 0.523 – 0.372 = 0.872 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objective 2 Compute probabilities by using the Addition Rule for Mutually Exclusive Events Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Mutually Exclusive Events Two events are said to be mutually exclusive if it is impossible for both events to occur. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example A die is rolled. Event A is that the die comes up 3, and event B is that the die comes up an even number. These events are mutually exclusive since the die cannot both come up 3 and come up an even number. A fair coin is tossed twice. Event A is that one of the tosses is heads, and Event B is that one of the tosses is tails. These events are not mutually exclusive since, if the two tosses are HT or TH, then both events occur. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The Addition Rule for Mutually Exclusive Events If events A and B are mutually exclusive, then P(A and B) = 0. This leads to a simplification of the General Addition Rule. If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B) Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example In the 2008 Olympic Games, a total of 11,028 athletes participated. Of these, 596 represented the United States, 332 represented Canada, and 85 represented Mexico. What is the probability that an Olympic athlete chosen at random represents the U.S. or Canada? Solution: These events are mutually exclusive, because it is impossible to compete for both the U.S. and Canada. So, P(U.S. or Canada) = P(U.S.) + P(Canada) = 596/11,028 + 332/11,028 = 928/11,028 = 0.08415 Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Objective 3 Compute probabilities by using the Rule of Complements Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Complements If there is a 60% chance of rain today, then there is a 40% chance that it will not rain. The events “Rain” and “No rain” are complements. The complement of an event says that the event does not occur. If A is any event, the complement of A is the event that A does not occur. The complement of A is denoted Ac. The Rule of Complements states that P(Ac) = 1 – P(A). Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Example According to the Wall Street Journal, 42% of cars sold in May 2008 were small cars. What is the probability that a randomly chosen car sold in May 2008 is not a small car? Solution: The events of choosing a small car and not choosing a small car are complements. P(Not a small car) = 1 – P(Small car) = 1 – 0.42 = 0.58. Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Do You Know… • • • How to compute probabilities using the General Addition Rule? How to compute probabilities of mutually exclusive events? How to use the Complement Rule to find probabilities? Copyright ©2014 The McGraw-Hill Companies, Inc. Permission required for reproduction or display.