Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Chapter Probability 5

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Section The Addition Rule and Complements 5.2

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objectives 1.Use the Addition Rule for Disjoint Events 2.Use the General Addition Rule 3.Compute the probability of an event using the Complement Rule 5-3

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 1 Use the Addition Rule for Disjoint Events 5-4

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events. 5-5

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. We often draw pictures of events using Venn diagrams. These pictures represent events as circles enclosed in a rectangle. The rectangle represents the sample space, and each circle represents an event. For example, suppose we randomly select a chip from a bag where each chip in the bag is labeled 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Let E represent the event “choose a number less than or equal to 2,” and let F represent the event “choose a number greater than or equal to 8.” These events are disjoint as shown in the figure. 5-6

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 5-7

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Addition Rule for Disjoint Events If E and F are disjoint (or mutually exclusive) events, then Addition Rule for Disjoint Events If E and F are disjoint (or mutually exclusive) events, then 5-8

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. The Addition Rule for Disjoint Events can be extended to more than two disjoint events. In general, if E, F, G,... each have no outcomes in common (they are pairwise disjoint), then 5-9

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. The probability model to the right shows the distribution of the number of rooms in housing units in the United States. Number of Rooms in Housing Unit Probability One0.010 Two0.032 Three0.093 Four0.176 Five0.219 Six0.189 Seven0.122 Eight0.079 Nine or more0.080 EXAMPLE The Addition Rule for Disjoint Events (a) Verify that this is a probability model. All probabilities are between 0 and 1, inclusive … =

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Number of Rooms in Housing Unit Probability One0.010 Two0.032 Three0.093 Four0.176 Five0.219 Six0.189 Seven0.122 Eight0.079 Nine or more0.080 EXAMPLE The Addition Rule for Disjoint Events 5-11 (b) What is the probability a randomly selected housing unit has two or three rooms? P(two or three) = P(two) + P(three) = = 0.125

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Number of Rooms in Housing Unit Probability One0.010 Two0.032 Three0.093 Four0.176 Five0.219 Six0.189 Seven0.122 Eight0.079 Nine or more0.080 EXAMPLE The Addition Rule for Disjoint Events 5-12 (c) What is the probability a randomly selected housing unit has one or two or three rooms? P(one or two or three) = P(one) + P(two) + P(three) = = 0.135

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 2 Use the General Addition Rule 5-13

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. The General Addition Rule For any two events E and F, The General Addition Rule For any two events E and F, 5-14

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Suppose that a pair of dice are thrown. Let E = “the first die is a two” and let F = “the sum of the dice is less than or equal to 5”. Find P(E or F) using the General Addition Rule. EXAMPLEIllustrating the General Addition Rule 5-15

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. 5-16

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Objective 3 Compute the Probability of an Event Using the Complement Rule 5-17

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Complement of an Event Let S denote the sample space of a probability experiment and let E denote an event. The complement of E, denoted E C, is all outcomes in the sample space S that are not outcomes in the event E. 5-18

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. Complement Rule If E represents any event and E C represents the complement of E, then P(E C ) = 1 – P(E) 5-19 Entire region The area outside the circle represents E c

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. According to the American Veterinary Medical Association, 31.6% of American households own a dog. What is the probability that a randomly selected household does not own a dog? P(do not own a dog) = 1 – P(own a dog) = 1 – = EXAMPLE Illustrating the Complement Rule 5-20

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. The data to the right represent the travel time to work for residents of Hartford County, CT. (a) What is the probability a randomly selected resident has a travel time of 90 or more minutes? EXAMPLE Computing Probabilities Using Complements Source: United States Census Bureau 5-21

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. EXAMPLE Computing Probabilities Using Complements Source: United States Census Bureau There are a total of 24, ,112 + … + 4,895 = 318,800 residents in Hartford County The probability a randomly selected resident will have a commute time of “90 or more minutes” is 5-22

Copyright © 2013, 2010 and 2007 Pearson Education, Inc. (b) Compute the probability that a randomly selected resident of Hartford County, CT will have a commute time less than 90 minutes. P(less than 90 minutes) = 1 – P(90 minutes or more) = 1 – =