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Chapter 11 Probability Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 1 Chapter 11: Probability 11.1 11.2 11.3 11.4 11.5 Basic Concepts Events Involving “Not” and “Or” Conditional Probability and Events Involving “And” Binomial Probability Expected Value and Simulation Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 2 Section 11-2 Events Involving “Not” and “Or” Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 3 Events Involving “Not” and “Or” • Know that the probability of an event is a real number between 0 and 1, inclusive of both, and know the meanings of the terms impossible event and certain event. • Understand the correspondences among set theory, logic, and arithmetic. • Determine the probability of “not E” given the probability of E. • Determine the probability of “A or B” given the probabilities of A, B, and A and B. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 4 Properties of Probability Let E be an event from the sample space S. That is, E is a subset of S. Then the following properties hold. 1. 0 P( E ) 1 (The probability of an event is between 0 and 1, inclusive.) 2. P () 0 (The probability of an impossible event is 0.) 3. P ( S ) 1 (The probability of a certain event is 1.) Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 5 Example: Finding Probability When Rolling a Die When a single fair die is rolled, find the probability of each event. a) the number 3 is rolled b) a number other than 3 is rolled c) the number 7 is rolled d) a number less than 7 is rolled Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 6 Example: Finding Probability When Rolling a Die Solution There are six possible outcomes for the die: {1, 2, 3, 4, 5, 6}. 1 a) the number 3 is rolled P (3) 6 5 P(not 3) 6 b) a number other than 3 is rolled c) the number 7 is rolled P (7) 0 d) a number less than 7 is rolled Copyright © 2016, 2012, and 2008 Pearson Education, Inc. P(less than 7) 1 7 Events Involving “Not” The table on the next slide shows the correspondences that are the basis for the probability rules developed in this section. For example, the probability of an event not happening involves the complement and subtraction. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 8 Correspondences Set Theory Logic Arithmetic Operation or Connective (Symbol) Complement Not Subtraction ( ) ( ) () Operation or Connective (Symbol) Union Or Addition ( ) () () Operation or Connective (Symbol) Intersection And Multiplication ( ) () () Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 9 Probability of a Complement The probability that an event E will not occur is equal to one minus the probability that it will occur. E S E P(not E ) P( S ) P( E ) 1 P( E ) So we have P( E ) P E 1 and P( E ) 1 P( E). Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 10 Example: Finding the Probability from a Complement When a single card is drawn from a standard 52-card deck, what is the probability that it will not be an ace? Solution P(not an ace) 1 P(ace) 4 1 52 48 12 . 52 13 Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 11 Events Involving “Or” Probability of one event or another should involve the union and addition. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 12 Mutually Exclusive Events Two events A and B are mutually exclusive events if they have no outcomes in common. (Mutually exclusive events cannot occur simultaneously.) Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 13 Addition Rule of Probability (for A or B) If A and B are any two events, then P( A or B) P( A) P( B) P( A and B). If A and B are mutually exclusive, then P( A or B) P( A) P( B). Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 14 Example: Finding the Probability of an Event Involving “Or” When a single card is drawn from a standard 52-card deck, what is the probability that it will be a king or a diamond? Solution P(king or diamond) P(K) P(D) P(K and D) 4 13 1 52 52 52 16 4 52 13 Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 15 Example: Finding the Probability of an Event Involving “Or” If a single die is rolled, what is the probability of a 2 or odd? Solution These are mutually exclusive events. P(2 or odd) P(2) P(odd) 1 3 4 2 . 6 6 6 3 Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 16 Example: Finding the Probability of an Event Involving “Or” Of 20 elective courses, Emily plans to enroll in one, which she will choose by throwing a dart at the schedule of courses. If 8 of the courses are recreational, 9 are interesting, and 3 are both recreational and interesting, find the probability that the course she chooses will have at least one of these two attributes. Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 17 Example: Finding the Probability of an Event Involving “Or” Solution If R denotes “recreational” and I denotes “interesting,” then 8 9 3 P( R) , P( I ) , P( R and I) = 20 20 20 R and I are not mutually exclusive. 8 9 3 P( R or I ) 20 20 20 14 7 20 10 Copyright © 2016, 2012, and 2008 Pearson Education, Inc. 18

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