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Section 2.1-2.2 Sample space (p.38) 2.4 (a) An experiment involves tossing a pair of dice, 1 green and 1 red, and recording the numbers that come up. If x equals the outcome on the green die and y the outcome on the red die, describe the sample space S by listing the elements (x, y). sol) S = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6), (5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6), (6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)} 2.5 An experiment consists of tossing a die and then flipping a coin once if the number on the die is even. If the number on the die is odd, the coin is flipped twice. Using the notation 4H, for example, to denote the outcome that the die comes up 4 and then the coin comes up heads, and 3HT to denote the outcome that the die comes up 3 followed by a head and then a tail on the coin, construct a tree diagram to show the 18 elements of the sample space S. sol) S = {1HH, 1HT, 1T H, 1T T, 2H, 2T, 3HH, 3HT, 3T H, 3T T, 4H, 4T, 5HH, 5HT, 5T H, 5T T, 6H, 6T } 1 2.14 If S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {0, 2, 4, 6, 8}, B = {1, 3, 5, 7, 9}, C = {2, 3, 4, 5}, and D = {1, 6, 7}, list the elements of the sets corresponding to the following events: (a) A ∪ C sol) A ∪ C = {0, 2, 3, 4, 5, 6, 8} (b) A ∩ B sol) A∩B =∅ (c) C 0 sol) C 0 = {0, 1, 6, 7, 8, 9} (d) (C 0 ∩ D) ∪ B sol) (C 0 ∩ D) ∪ B = {1, 6, 7} ∪ {1, 3, 5, 7, 9} = {1, 3, 5, 6, 7, 9} 0 (e) (S ∩ C) sol) 0 (S ∩ C) = C 0 = {0, 1, 6, 7, 8, 9} (f) A ∩ C ∩ D0 sol) A ∩ C ∩ D0 = {0, 2, 4, 6, 8} ∩ {2, 3, 4, 5} ∩ {0, 2, 3, 4, 5, 8, 9} = {2, 4} 2.17 Let A , B , and C be events relative to the sample space S. Using Venn diagrams, shade the areas representing the following events: 0 (a) (A ∩ B) sol) 2 0 (b) (A ∪ B) sol) (c) (A ∩ C) ∪ B sol) Section 2.3 Counting (p.47) 2.22 In a medical study patients are classified in 8 ways according to whether they have blood type AB + , AB − , A+ , A− , B + , B − , O+ , or O− , and also according to whether their blood pressure is low, normal, or high. Find the number of ways in which a patient can be classified. sol) With n1 = 8 blood types and n2 = 3 classifications of blood pressure, the multiplication rule gives n1 n2 = (8)(3) = 24 classifications. 2.30 In how many different ways can a true-false test consisting of 9 questions be answered? sol) n1 = 2, n2 = 2, n3 = 2, n4 = 2, n5 = 2, n6 = 2, n7 = 2, n8 = 2, n9 = 2 By theorem 2.1, n1 n2 n3 n4 n5 n6 n7 n8 n9 = (2)(2)(2)(2)(2)(2)(2)(2)(2) = 29 = 512 possible ways. 2.31 If a multiple-choice test consists of 5 questions each with 4 possible answers of which only 1 is correct, (a) In how many different ways can a student check off one answer to each question? sol) 3 n1 = 4, n2 = 4, n3 = 4, n4 = 4, n5 = 4 By theorem 2.1, n1 n2 n3 n4 n5 = (4)(4)(4)(4)(4) = 45 = 1024 possible ways. (b) In how many ways can a student check off one answer to each question and get all the answers wrong? sol) n1 = 3, n2 = 3, n3 = 3, n4 = 3, n5 = 3 By theorem 2.1, n1 n2 n3 n4 n5 = (3)(3)(3)(3)(3) = 35 = 243 possible ways. 2.33 A witness to a hit-and-run accident told the police that the license number contained the letters RLH followed by 3digits, the first of which is a 5. If the witness cannot recall the last 2 digits, but is certain that all 3 digits are different, find the maximum number of automobile registrations that the police may have to check. sol) n1 = 9 (since all possible numbers are 0, 1, 2, 3, 4, 6, 7, 8, 9), n2 = 8 (= n1 − 1) By theorem 2.1, n1 n2 = (9)(8) = 72 2.37 In how many ways can 4 boys and 5 girls sit in a row if the boys and girls must alternate? sol) G indicates a girl’s sit. B indicates a boy’s sit. GBGBGBGBG Girls can sit 5! different ways and boys can sit 4! different ways. Thus, 5! × 4! = 2880 possible ways. 4