1. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 12.5 Tree Diagrams

2. Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Counting Principle Tree Diagrams 12.5-2

3. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Counting Principle If a first experiment can be performed in M distinct ways and a second experiment can be performed in N distinct ways, then the two experiments in that specific order can be performed in M N distinct ways. 12.5-3

4. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Definitions Sample space: A list of all possible outcomes of an experiment. Sample point: Each individual outcome in the sample space. Tree diagrams are helpful in determining sample spaces. 12.5-4

5. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Selecting Balls without Replacement Two balls are to be selected without replacement from a bag that contains one red, one blue, one green and one orange ball. a)Use the counting principle to determine the number of points in the sample space. Solution There are 4 3 = 12 sample points. 12.5-5

6. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Selecting Balls without Replacement b)Construct a tree diagram and list the sample space. Solution The first ball selected can be red, blue, green, or orange. Since this experiment is done without replacement, the same colored ball cannot be selected twice. See the tree diagram on the next slide. 12.5-6

7. Copyright 2013, 2010, 2007, Pearson, Education, Inc. 12.5-7

8. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Selecting Balls without Replacement c)Find the probability that one orange ball is selected. Solution 12 possible outcomes 6 have one orange ball: RO, BO, GO, OR, OB, OG 12.5-8

9. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: Selecting Balls without Replacement d)Find the probability that a green ball followed by a red ball is selected. Solution One outcome meets this criteria: GR 12.5-9

10. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Selecting Ticket Winners A radio station has two tickets to give away to a Bon Jovi concert. It held a contest and narrowed the possible recipients down to four people: Christine (C), Mike Hammer (MH), Mike Levine (ML), and Phyllis (P). The names of two of these four people will be selected at random from a hat, and the two people selected will be awarded the tickets. 12.5- 10

11. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Selecting Ticket Winners a)Use the counting principle to determine the number of points in the sample space. Solution There are 4 3 = 12 sample points in the sample space. 12.5- 11

12. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Selecting Ticket Winners b)Construct a tree diagram and list the sample space. Solution See tree diagram on the next slide. 12.5- 12

13. Copyright 2013, 2010, 2007, Pearson, Education, Inc. b)Construct a tree diagram and list the sample space. 12.5- 13

14. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Selecting Ticket Winners c)Determine the probability that Christine is selected. Solution 12 possible outcomes 6 have Christine: C MH, C ML, C P, MH C, ML C, P C 12.5- 14

15. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Selecting Ticket Winners d)Determine the probability that neither Mike Hammer nor Mike Levine is selected. Solution 2 outcomes have neither Mike: C P, P C 12.5- 15

16. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 3: Selecting Ticket Winners e)Determine the probability that at least one Mike is selected. Solution There are 10 outcomes with at least one Mike; all those except C P and P C. 12.5- 16

17. Copyright 2013, 2010, 2007, Pearson, Education, Inc. P(event happening at least once) 12.5- 17

18. Copyright 2013, 2010, 2007, Pearson, Education, Inc. P(event happening at least once) In part d) of Example 3, we found that In part e) we could have used 12.5- 18