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Geometry 11.8 Geometric Probability

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Ways to Measure Probability LikelihoodProbability as a Fraction Probability as a Decimal Probability as a % Certain No Chance Good Chance 8/8 = 1/1 = 11.0 = 1 100/100 = 100% 0/8 = 00.0 = 0 0/100 = 0% 6/8 = 3/40.75 75/100 = 75%

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This lesson will use two basic principles. The first principle involves the lengths of segments. Suppose a point P of XY is picked at random. Then: probability that P is on X Q Y

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Solve. A B C D GF E 1) A point X is picked at random on. What is the probability that X is on: a. b. c. d. e. f. 13 spaces 4 spaces 4/13 2 spaces 2/13 6 spaces 6/13 7 spaces 7/13 15 spaces It is certain. 1 2 spaces (not between A and F) 0

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2. M is the midpoint of, Q is the midpoint of, and X is the midpoint of. If a point on JK is picked at random, what is the probability that the point is on ?. X QJ M K The probability is ½. ½

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3) Every 20 minutes a bus pulls up outside a busy department store and waits for five minutes while passengers get on and off. Then the bus leaves. If a person walks out of the department store at a random time, what is the probability that a bus is there? Think of it as a timeline! 0 min.20 min. Bus Arrives Bus Departs One Full Cycle 5 min. 5 minutes/20 minutes The probability that a bus is there is ¼.

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4) A piece of rope 20 ft long is cut into two pieces at a random point. What is the probability that both pieces of rope will be at least 3 ft long? 20 feet 3 feet Do not cut or one piece will be less than 3 feet! 3 feet Do not cut! 14 feet of rope to cut! The probability of cutting the rope so both pieces will be at least 3 feet is 70%. 14/20 = 7/10

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The second principle of geometric probability involves areas of regions. Suppose a point P of region A is picked at random. Then: B A A B probability that P is in region B =

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If a value of pi is required in the following exercises, use π = 3.14. 5. A dart lands at a random point on the square dartboard shown. 5 10 25 5 10 a) What is the probability that the dart lands within the larger circle? b) What is the probability that the dart lands within the smaller circle? Probability = Area of smaller circle Area of square = 3.14(5) 2 25 2 = 78.5 625 =0.1256 = 12.56% Probability = Area of larger circle Area of square = 3.14(10) 2 25 2 = 314 625 =0.5024 = 50.24%

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6. A circular archery target has diameter 60 cm. Its bull’s eye has diameter 10 cm. a)If an amateur shoots an arrow and the arrow hits a random point on the target, what is the probability that the arrow hits the bull’s eye? b)After many shots, 100 arrows have hit the target. Estimate the number hitting the bull’s eye. 30 5 Probability = Area of bull’s eye Area of target = 3.14(5) 2 3.14(30) 2 = 78.5 2826 =.0278 = 2.78% Using the answer of about 3% from above, approximately 3 of 100 would hit the target.

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7. A dart is thrown at a board 200 cm long and 157 cm wide. Attached to the board are 10 balloons, each with radius 12 cm. Assuming that each balloon lies entirely on the board, what is the probability that a dart that hits the board also hits a balloon. 200 157 12 Probability = 10(Area of one balloon) Area of board = 10(3.14)(12) 2 200(157) = 4521.6 31400 =0.144 = 14.4%

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8. A ship has sunk in the ocean in a square region 5 miles on a side. A salvage vessel anchors at a random spot in this square. Divers search half a mile in all directions from the point on the ocean floor directly below the vessel. a) What is the approximate probability that they locate the sunken ship at the first place they anchor? b)What is the approximate probability that they locate the sunken ship in five tries? (Assume that the tries do not overlap.) 5 miles ½ Probability = Search Area Total Area = 3.14( ½ ) 2 (5) 2 = 0.785 25 =.0314 = 3.14% Multiply the last answer by 5 to get 15.7%

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HOMEWORK pg. 462 CE #1-4 WE #1-8

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