696 CHAPTER 11 Probability For example, if you are planting flowers, suppose that the probability of not getting any red flowers from the seeds that are planted is . 2 7 Then the probability of getting at least one red flower from the seeds that are planted is − = 1 . 2 7 5 7 We will use this rule in later sections. In all the tree diagrams in this section, the outcomes were always equally likely; that is, each outcome had the same probability of occurrence. Consider a rock that has 4 faces such that each face has a different surface area and the rock is not uniform in density (see Fig. 11.10). When the rock is dropped, the probability that the rock lands on face 1 will not be the same as the probability that the rock lands on face 2. In fact, the probabilities that the rock lands on face 1, face 2, face 3, and face 4 may all be different. Therefore, the outcomes of the rock landing on face 1, face 2, face 3, and face 4 are not equally likely outcomes. Because the outcomes are not equally likely and we are not given additional information, we cannot determine the theoretical probability of the rock landing on each individual face. However, we can still determine the sample space indicating the faces that the rock may land on when the rock is dropped twice. The tree diagram and sample space are shown in Fig. 11.11. Figure 11.10 Sample Space First Drop Second Drop 1 2 3 4 11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 Figure 11.11 Since the outcomes are not equally likely, the probability of each of the 16 sample points in the sample space occurring cannot be determined. If the outcomes were equally likely, then each of the 16 points in the sample space would have a probability of . 1 16 See Exercises 27 and 28, which deal with outcomes that are not equally likely. Instructor Resources for Section 11.4 in MyLab Math • Objective-Level Videos 11.4 • StatCrunch: Simulation: Urn Sampling • PowerPoint Lecture Slides 11.4 • MyLab Exercises and Assignments 11.4 Exercises Warm Up Exercises In Exercises 1– 4, fill in the blank with an appropriate word, phrase, or symbol(s). 1. A list of all possible outcomes of an experiment is called a(n) ___________ space. Sample 2. Each individual outcome in a sample space is called a sample ____ _______ . Point 3. If a first experiment has 3 distinct outcomes and a second experiment has 8 distinct outcomes, then the two experiments in that specific order have ___________ distinct outcomes. 24 4. A helpful method to determine a sample space is to construct a(n) ___________ diagram. Tree Practice the Skills 5. Scrabble Tiles A bag contains 26 Scrabble tiles—each with a different letter of the alphabet. If two tiles are randomly drawn from the bag, use the fundamental counting principle to determine the number of outcomes in the sample space if the tiles are selected a) with replacement. 676 b) without replacement. 650 6. Apples to Apples Cards In the game Apples to Apples, the green apple card deck contains 108 cards. If two cards are randomly dealt from this deck, use the fundamental SECTION 11.4
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