Geometry

1. Geometry 11.8 Geometric Probability

2. Ways to Measure Probability Certain 8/8 = 1/1 = 1 1.0 = 1 100/100 = 100% No Chance 0/8 = 0 0.0 = 0 0/100 = 0% Good Chance 6/8 = 3/4 0.75 75/100 = 75%

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

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

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

6. 5 25 10 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. 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? 5 Area of larger circle 3.14(10)2 314 10 Probability = = = = 0.5024 = 50.24% Area of square 252 625 Area of smaller circle 3.14(5)2 78.5 Probability = = = = 0.1256 = 12.56% Area of square 252 625

7. Push your rows together and complete two of #2-4 and two of #6-8.

8. J Q M X K 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 ? ½ The probability is ½ . .

9. 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? 5 minutes/20 minutes Think of it as a timeline! The probability that a bus is there is ¼ . One Full Cycle 0 min. 20 min. 5 min. Bus Arrives Bus Departs

10. 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? 14/20 = 7/10 14 feet of rope to cut! 3 feet Do not cut or one piece will be less than 3 feet! 3 feet Do not cut! 20 feet The probability of cutting the rope so both pieces will be at least 3 feet is 70%.

11. 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? 6. A circular archery target has diameter 60 cm. Its bull’s eye has diameter 10 cm. 30 Area of bull’s eye 3.14(5)2 78.5 5 Probability = = = = .0278 = 2.78% Area of target 3.14(30)2 2826 • After many shots, 100 arrows have hit the target. Estimate the number • hitting the bull’s eye. Using the answer of about 3% from above, approximately 3 of 100 would hit the target.

12. 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. 12 12 12 12 12 12 157 12 12 12 12 200 10(Area of one balloon) 10(3.14)(12)2 4521.6 Probability = = = = 0.144 = 14.4% Area of board 200(157) 31400

13. 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. • What is the approximate probability that they locate the sunken ship at the • first place they anchor? • What is the approximate probability that they locate the sunken ship in five tries? • (Assume that the tries do not overlap.) 5 miles ½ Search Area 3.14( ½ )2 0.785 Probability = = = = .0314 = 3.14% Total Area (5)2 5 miles 25 Multiply the last answer by 5 to get 15.7%

14. HW • P. 462-463 (CE 1-4 WE 1-8) • Ch 11 Test on Friday • Review Tomorrow and Thursday