1. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 1

2. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 15 Probability Rules!

3. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 3 The General Addition Rule When two events A and B are disjoint, we can use the addition rule for disjoint events from Chapter 14: P(A or B) = P(A) + P(B) However, when our events are not disjoint, this earlier addition rule will double count the probability of both A and B occurring. Thus, we need the General Addition Rule.

4. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 4 The General Addition Rule (cont.) General Addition Rule: For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B) The following Venn diagram shows a situation in which we would use the general addition rule:

5. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 5 The General Addition Rule (cont.) Data suggest that 65% of new laptop computers have a DVD drive, 75% are wireless and 47% have both features. What is the probability that a newly purchased computer has a DVD drive or is wireless? does not have a DVD drive and is not wireless? is wireless but does not have a DVD drive?

6. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 6 It Depends… When we want the probability of an event from a conditional distribution, we write P(B|A) and pronounce it “the probability of B given A.” A probability that takes into account a given condition is called a conditional probability.

7. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 7 A pair of dice is thrown one at a time. Let A be the event that the sum of 9 is rolled. Let B be the event that the first die thrown is a 2. Let C be the event that the first die thrown is a 5. Let D be the event that the sum of 7 is rolled. i. What is the probability the sum of the dice is 9? ii. What is the probability the sum of the dice is 9, given that the first die rolled is 2? iii. What is the probability the sum of the dice is 9, given that the first die rolled is 5? iv. What is the probability the sum of the dice is 7? v. What is the probability the sum of the dice is 7, given that the first die rolled is 2? vi. What is the probability the sum of the dice is 7, given that the first die rolled is 5?

8. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 8 It Depends… (cont.) To find the probability of the event B given the event A, we restrict our attention to the outcomes in A. We then find the fraction of those outcomes B that also occurred. Note: P(A) cannot equal 0, since we know that A has occurred.

9. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 9 A study at a local bar found people of various ages playing games. Find the probability that a randomly selected person Plays darts. Is 21-29. Is 21-29 given that they are playing darts. Is 21-29 given that they are singing karaoke. Is singing karaoke given that they are 21-29. Is 30-39 and playing pool. Is playing pool given that they are 30-39. 21-2930-3940-4950 and olderTotal Darts41215637 Pool817161152 Karaoke1750123 Total29343118112

10. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 10 Find the Probability that a randomly selected student drinks and smokes drinks given that they smoke smokes given that they drink Smoke YesNo DrinkYes.23.20 No.09.48

11. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 11 The General Multiplication Rule When two events A and B are independent, we can use the multiplication rule for independent events from Chapter 14: P(A and B) = P(A) x P(B) However, when our events are not independent, this earlier multiplication rule does not work. Thus, we need the General Multiplication Rule.

12. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 12 The General Multiplication Rule (cont.) We encountered the general multiplication rule in the form of conditional probability. Rearranging the equation in the definition for conditional probability, we get the General Multiplication Rule: For any two events A and B, P(A and B) = P(A) x P(B|A) or P(A and B) = P(B) x P(A|B)

13. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 15- 13 The General Multiplication Rule (cont.) An box contains 3 white balls, 4 red balls and 5 black balls. A ball is picked, its color recorded but is not returned to the box. Another ball is then selected and its color recorded. Find the probability that 2 black balls are selected. Find the probability that 2 balls of the same color are selected.