Lial/Hungerford/Holcomb/Mullins: Mathematics with Applications 11e Finite Mathematics with Applications 11e Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Chapter 8 Sets and Probability Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Section 8.1 Sets Copyright ©2015 Pearson Education, Inc. All right reserved.

List the elements belonging to each of the given sets. Example: List the elements belonging to each of the given sets. (a) Solution: The natural numbers less than 5 make up the set {1, 2, 3, 4}. (b) Solution: The states that border Florida make up the set {Alabama, Georgia}. Copyright ©2015 Pearson Education, Inc. All right reserved.

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Applications of Venn Diagrams Section 8.2 Applications of Venn Diagrams Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Use the following Venn diagram with two overlapping circles, and shade the regions representing the given sets. (a) Solution: Set contains all the elements outside set A. As labeled in the Venn diagram above, the set is represented by regions 1 and 4. Set B is represented by the elements in regions 3 and 4. The intersection of sets is given by the region common to regions 1 and 4 and regions 3 and 4. The result, region 4, is shaded below. Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Use the following Venn diagram with two overlapping circles, and shade the regions representing the given sets. (b) Solution: Again, set is represented by regions 1 and 4 and set by regions 1 and 2. To find identify the region that represents the set of all elements in The result includes regions 1, 2, and 4, which are shaded below. Copyright ©2015 Pearson Education, Inc. All right reserved.

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Introduction to Probability Section 8.3 Introduction to Probability Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Example: Suppose a fair die is rolled. Then the sample space is {1, 2, 3, 4, 5, 6}. Find the requested events. (a) The event “the die shows a 4.” Solution: {4}. (b) The event “the number showing is less than 10.” Solution: The event is the entire sample space {1, 2, 3, 4, 5, 6}. This event is a certain event; if a die is rolled, the number showing (either 1, 2, 3, 4, 5, or 6) must be less than 10. (c) The event “the die shows a 7.” Solution: The empty set, this is an impossible event. Copyright ©2015 Pearson Education, Inc. All right reserved.

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Basic Concepts of Probability Section 8.4 Basic Concepts of Probability Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Example: If a fair die is rolled, what is the probability that any number but 5 will come up? Solution: If E is the event that 5 comes up, then is the event that any number but 5 comes up. Copyright ©2015 Pearson Education, Inc. All right reserved.

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Conditional Probability and Independent Events Section 8.5 Conditional Probability and Independent Events Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. Among partnerships and corporations that file tax returns in the United States, the probability that the enterprise consists of a firm dedicated to real estate, rentals, or leasing is .241. The probability that the firm is a partnership is .352. The probability that a firm is dedicated to real estate, rentals, or leasing or is a partnership is .424. Are the events of a firm being dedicated to real estate, rentals, or leasing and a firm being a partnership independent? Example: Solution: Let E represent the event the firm is dedicated to real estate, rentals, or leasing and let F represent the event the firm is a partnership. We must determine whether We know that Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Among partnerships and corporations that file tax returns in the United States, the probability that the enterprise consists of a firm dedicated to real estate, rentals, or leasing is .241. The probability that the firm is a partnership is .352. The probability that a firm is dedicated to real, estate rentals, or leasing or is a partnership is .424. Are the events of a firm being dedicated to real estate, rentals, or leasing and a firm being a partnership independent? Example: Solution: By the addition rule, we know that Therefore, Since these two events are not independent. Copyright ©2015 Pearson Education, Inc. All right reserved.

Copyright ©2015 Pearson Education, Inc. All right reserved. Section 8.6 Bayes’ Formula Copyright ©2015 Pearson Education, Inc. All right reserved.

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Copyright ©2015 Pearson Education, Inc. All right reserved. A manufacturer buys items from six different suppliers. The fraction of the total number of items obtained from each supplier, along with the probability that an item purchased from that supplier is defective, is shown in the following table: Example: Find the probability that a defective item came from supplier 5. Solution: Let F1 be the event that an item came from supplier 1, with defined in a similar manner. Let E be the event that an item is defective. We want to find By Bayes’ formula, There is about a 32% chance that a defective item came from supplier 5. Even though the supplier has only 3% defectives, his probability of being “guilty” is relatively high, about 32%, because of the large fraction of items he supplies. Copyright ©2015 Pearson Education, Inc. All right reserved.