692 CHAPTER 11 Probability Tree Diagrams A list of all the possible outcomes of an experiment is called a sample space. Each individual outcome in the sample space is called a sample point. A tree diagram is a diagram that displays all the sample points in the sample space. Tree diagrams are helpful tools when solving probability problems. A tree diagram illustrating all the possible outcomes when a coin is tossed and a die is rolled (see Fig. 11.5) has two initial branches, one for each possible outcome of the coin. Each of these branches will have six branches emerging from them, one for each possible outcome of the die. That will give a total of 12 branches, the same number of possible outcomes determined by using the fundamental counting principle. We can obtain the sample space by listing all the possible combinations of branches. Note that this sample space consists of 12 sample points. Thus, there are 2704 ways to draw two cards with replacement from a standard deck of cards. b) Without replacement means that after the first card is drawn, it is not replaced in the deck. The first card drawn can be any one of the 52 cards in the deck. Since the first card is not replaced in the deck, the second card can be any one of the 51 cards remaining in the deck. Using the fundamental counting principle, we have the following. = ⋅ = without Number of ways to draw two cards replacement 52 51 2652 Thus, there are 2652 ways to draw two cards without replacement from a standard deck of cards. 7 Now try Exercise 5 Possible Outcomes from Coin Possible Outcomes from Die Sample Space 1 6 5 4 3 2 H1 H6 H5 H4 H3 H2 1 6 5 4 3 2 T1 T6 T5 T4 T3 T2 H T Figure 11.5 Example 2 Selecting Ticket Winners A radio station has two tickets to give away to a comic book convention. It held a contest and narrowed the possible recipients down to four people: Christine (C), Mike (M), Larry (L), and Phyllis (P). The names of two of these four people will be randomly selected from a hat, and the two people selected will be awarded the tickets. a) Determine the number of sample points in the sample space. b) Construct a tree diagram and list the sample space. c) Determine the probability that Christine is selected. d) Determine the probability that Christine is selected and then Mike is selected.
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