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Independent Events OBJ: Find the probability of independent events
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DEF: Independent Events 2 events that do not depend on each other (one event occurring has no relationship to the other event occurring)
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EX: In a throw of a red die, r, and a white die, w, find: P(r 3 and w 2) P (r ≤ 3 and w ≤ 2) P (r ≤ 3) 18 (r 3) 36 (die pairs) 1 2 P (w ≤ 2) 12 (w ≤ 2) 36 (die pairs) 1 3 P (r ≤ 3 and w ≤ 2) P (r 3 ∩ w ≤ 2) 1 2 3 1 6 (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)
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EX: In a throw of a red die, r, and a white die, w, find: P (r=2 or w 5) P (r = 2 or w 5) P (r = 2) 6 1 (r = 2) 36 6 (die pairs) P (w 5) 12 1 (w 5) 36 3 (die pairs) P (r = 2 and w 5) P (r = 2 ∩ w 5) 2 1 36 18 P (r =2 or w 5) P (r = 2) + P (w 5) – P (r = 2 ∩ w 5) 6 + 12 – 2 36 36 36 16 36 4 9 (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)
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EX: In a throw of a red die, r, and a white die, w, find: P(r 2 and w 4) P (r 2 and w 4) P (r ≤ 2) 12 (r 2) 36 (die pairs) 1 3 P (w ≤ 4) 24 (w ≤ 4) 36 (die pairs) 2 3 P (r 2 and w 4) P (r 2 ∩ w 4) 1 2 3 3 2 9 (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)
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EX: In a throw of a red die, r, and a white die, w, find: P(r 4 and w=4) P(r 4 and w = 4) P(r 4) 18 r 4 36 (die pairs) 1 2 P (w = 4) 6 (w = 4) 36 (die pairs) 1 6 P(r 4 and w = 4) P(r 4 ∩ w = 4) 1 2 6 1 12 (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)
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EX: In a throw of a red die, r, and a white die, w, find:P(r 5 and w 2) P (r 5 and w 2) P (r 5 ) 12 (r 5 36 (die pairs) 1 3 P (w ≤ 2) 12 (w ≤ 2) 36 (die pairs) 1 3 P (r 5 and w 2) P (r 5 ∩ w 2) 1 3 1 9 (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)
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Make a sample space using a tree diagram showing all the possibilities for boys and girls in a family with three children. Girl (G) or Boy (B) G or B G or B G or B GGG GGB GBG GBB BGG BGB BBG BBB
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EX: Find: GGG, GGB, GBG, GBB, BGG, BGB, BBG, BBB P(3 boys) P(BBB) · 1 8 P( 2 boys and a girl) (BBG or BGB or GBB) · · + · · + · · 3 ( · · ) 3 8
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EX: Find: GGG, GGB, GBG, GBB, BGG, BGB, BBG, BBB 3) P (oldest child is a girl) (1 st 4 in sample space) 4 8 1 2 5) P (at most there is three boys) 8 8 1
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A lottery game consists of choosing a sequence of 3 digits. Repetition of digits is allowed, and any digit may be in any position. 6)P (654) __ __ __ 1010 10 1 1 1 1010 10 1_ 1000 7)P (no digit is zero) __ __ __ 1010 10 9 9 9 1010 10 729 1000
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Draw 3 cards, replacing after each draw. 12) P (only one red) RRR RRB RBR RBB BRR BRB BBR BBB 3 8 15) P (1 st card is 4) 4 52 52 52 52 52 1 13 16) P (only first card is a 4) __ __ __ 1313 13 1 12 12 13 13 13 144 2197
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A jar contains 5 blue marbles, 6 red marbles, and 4 yellow marbles. Draw 3 marbles, one at a time, replacing each marble after it’s drawn. 18.P( two blue and one red) RRR, RRB, RRB, RBB, BRR, BRB, BBR, BBB P (BBR, BRB, RBB) P(B) = 5 = 1 15 3 P(R) = 6 = 2 15 5 3 ( 1) (1) ( 2) 3 3 5 2 15
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A jar contains 5 blue marbles, 6 red marbles, and 4 yellow marbles. Draw 3 marbles, one at a time, replacing each marble after it’s drawn. 20. P (one of each color) 5 6 4 15 15 15 1 2 4 3 5 15 3 2 1 1 st 2 nd 3 rd color color color 6(1 2 4) 3 5 15 16 75
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