11.4 Tree Diagrams 693 The fundamental counting principle can be extended to any number of experiments, as illustrated in Example 3. Sample Space Second Selection M First Selection P L C P L C P M C CM CP CL MC MP ML LC LP LM PC PL PM L M L P M C Solution a) The first selection may be any one of the four people; see Fig. 11.6. Once the first person is selected, only three people remain for the second selection. Thus, there are ⋅ 4 3, or 12, sample points in the sample space. b) Now try Exercise 13 Figure 11.6 c) If we know the sample space, we can compute probabilities using the formula = P E E ( ) number of outcomes favorable to total number of outcomes The total number of outcomes will be the number of points in the sample space. From Fig. 11.6, we determine that there are 12 possible outcomes when two people are selected. Six of the outcomes have Christine. They are CM, CL, CP, MC, LC, PC. = = P(Christine is selected) 6 12 1 2 d) One possible outcome meets the criteria when Christine is selected and then Mike is selected: CM. = P(Christine is selected and then Mike is selected) 1 12 7 RECREATIONAL MATH JUMBLE You may have seen JUMBLE puzzles in newspapers, magazines, online, or with a phone app. In these puzzles the letters of a word are scrambled, and your goal is to unscramble the letters to make a word. For each JUMBLE below, (a) Use the fundamental counting principle to determine the number of possible arrangements of letters given. (b) Determine the word that results when the letters are placed in their proper order (note that only one word is possible). 1. KANCS 2. CYMITS The answers are listed upside down below. See Exercises 31 and 32 for more examples. We will expand upon this Recreational Math in Section 11.7. 1. a) 120 b) SNACK 2. a) 720 b) MYSTIC Example 3 Lunch Choices At Tops Diner, each lunch special consists of a sandwich, a salad, and a beverage. The sandwich choices are roast beef (R), ham (H), or turkey (T). The salad choices are macaroni (M) or potato (P). The beverage choices are coffee (C) or soda (S). a) Determine the number of different lunch specials offered by this restaurant. b) Construct a tree diagram and list the sample space. c) If a customer randomly selects one of the lunch specials, determine the probability that a roast beef sandwich and soda are selected. d) If a customer randomly selects one of the lunch specials, determine the probability that neither macaroni salad nor coffee is selected. Solution a) There are 3 choices for a sandwich, 2 choices for a salad, and 2 choices for a beverage. Using the fundamental counting principle, we can determine that there are ⋅ ⋅ 3 2 2, or 12, different lunch specials.
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