Section 12.5 Tree Diagrams

What You Will Learn Counting Principle Tree Diagrams

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.

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.

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.

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.

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

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

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.

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.

Example 3: Selecting Ticket Winners b) Construct a tree diagram and list the sample space. Solution See tree diagram on the next slide.

b) Construct a tree diagram and list the sample space.

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

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

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.

P(event happening at least once)

P(event happening at least once) In part d) of Example 3, we found that In part e) we could have used