720 CHAPTER 11 Probability Sumiko works at an animal shelter that has seven dogs available for adoption. Sumiko is preparing a poster that will feature a photograph of each of the seven dogs. How many different arrangements of the seven photographs are possible? If one of these arrangements of photographs is randomly selected, what is the probability that the photographs are in alphabetical order by the dogs’ names? In this section, we will answer questions like these and discuss many other applications involving the number of ordered arrangements of a set of objects. SECTION 11.7 The Fundamental Counting Principle and Permutations LEARNING GOALS Upon completion of this section, you will be able to: 7 Solve problems using the fundamental counting principle. 7 Solve problems involving permutations. 7 Solve problems involving permutations of duplicate items. 73. a blend, given that it is a medium company stock. 15 52 74. a large company stock, given that it is a blend stock. 23 50 75. Venn Diagram Consider the following Venn diagram. The numbers in the regions of the circle indicate the number of items that belong to that region. For example, 60 items are in set A but not in set B. Assume one item is selected at random from the universal set. Determine 77. Given that P A P B () 0.3, () 0.5, = = and P A B ( and ) 0.15, = use the formula P E E P E E P E ( ) ( and ) ( ) 2 1 1 2 1 = to determine a) P A B ( ). 0.3 b) P B A ( ). 0.5 c) Are A and B independent? Explain. PAB PA PBA PB Yes; ( ) ( ) and ( ) ( ). = = Recreational Mathematics In Exercises 78–83, suppose that each circle is equally likely to be selected. One circle is randomly selected. + – + + – – Determine the probability indicated. 78. P(green circle obtained) + 1 3 79. P( orange circle obtained) + 1 2 80. P(yellow circle obtained) − 1 3 81. P(green obtained) + + 1 3 82. P(green or yellow circle green obtained) + 1 83. P(yellow circle with green obtained) + + 1 3 Research Activity 84. Bayes’ Theorem Read the Recreational Math box on page 715 regarding the Monty Hall problem. Do research and write a paper on Bayes’ theorem. Include a discussion of how Bayes’ theorem can be used to solve the Monty Hall problem. 60 80 40 20 U B A a) n A( ) b) n B( ) c) P A( ) d) P B( ) 140 120 7 10 3 5 Use the conditional probability formula to determine e) P A B ( ). 2 3 f) P B A ( ). 4 7 g) Explain why A and B are not independent events. Because P A B P A ( ) ( ) ≠ and P B A P B ( ) ( ). ≠ 76. A formula we gave for conditional probability is P E E n E E n E ( ) ( and ) ( ) 2 1 1 2 1 = This formula may be derived from the formula P E E P E E P E ( ) ( and ) ( ) 2 1 1 2 1 = Can you explain why? [Hint: Consider what happens to the denominators of P E E (and ) 1 2 and P E( ), 1 when they are expressed as fractions and the fractions are divided out.] The denominators of the probabilities will be the same and thus will divide out. L-N/Shutterstock
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