18.2 Geometric Probability

Transcription

1 Name Class Date 18.2 Geometric Probability Essential Question: What is geometric probability? Explore G.13.B Determine probabilities based on area to solve contextual problems. Using Geometric Probability to Estimate Pi Resource Locker Remember that in probability, the set of all possible outcomes of an experiment is called the sample space. Any set of outcomes is called an event. If every outcome in the sample space is equally likely, the theoretical probability of an event is: P = number of outcomes in the event number of outcomes in the sample space. Geometric probability is used when an experiment has an infinite number of outcomes. In geometric probability, the probability of an event is based on a ratio of geometric measures such as length or area. The outcomes of an experiment may be points on a segment or in a plane figure. You can use geometric probability to estimate π. The squares in the grid are the same width as the diameter of a penny: 0.75 in., or mm. Toss a penny onto the grid 20 times. Let x represent the number of times the penny lands touching or covering an intersection of two grid lines. Module Lesson 2

2 B Estimate π using the formula π 4 x_ 20. C How close is your result to π? Average the results of the entire class to get a more accurate estimate. Reflect 1. In order for a penny to touch or cover an intersection, the center of the penny can land anywhere in the shaded area of the sample grid square shown. So the area of the shaded region is the number of outcomes in the event. The area of the square is the number of outcomes in the sample space. Write a ratio to represent the geometric probability of the center of the penny landing in the shaded area. r 2. Explain why the formula in the activity can be used to estimate π. Explain 1 Using Length to Find Probability In the Explore, you applied geometric probability using an area model. Length and angle measure are two other models you can use. Geometric Probability Model Length Angle Measure Area Example Sample Space Event A B C D All points on _ AD All points in the circle All points in the rectangle All points on _ BC All points in the shaded region All points in the triangle Probability P = BC_ AD P = measure of angle P = area of triangle 360 area of rectangle Module Lesson 2

3 Example 1 Use length to find probability and solve problems. A stoplight has the following cycle: green for 25 seconds, yellow for 5 seconds, and red for 30 seconds. What is the probability that the light will be yellow when you arrive? If you arrive at the light 50 times, predict how many times you will have to stop and wait more than 10 seconds. To find the probability, draw a segment to represent the number of seconds that each color light is on. A Green Yellow Red 25 B C 30 5 D The light is yellow for 5 out of every 60 seconds. 5 So P = 60 = In the model, the event of stopping and waiting more than 10 seconds is represented by a segment that starts at C and ends 10 units from D. The probability of stopping and waiting more than 10 seconds is P = = 1_ 3. If you arrive at the light 50 times, you will probably stop and wait more than 10 seconds about 1 (50) 17 times. 3 B A bus comes to a station every 10 minutes and waits at the station for 1.5 minutes. Find the probability that a bus will be at the station when you arrive. If you go to the station 20 times, predict about how many times you will have to wait less than 3 minutes to board a bus. Buses arrive at the station 10 minutes apart. Suppose a bus arrives at time A and another bus arrives at time B. The bus that arrives at time A waits 1.5 minutes and departs at time C. Draw a segment relating A, B, and C. A bus is at the station for 1.5 out of every So, P = _ 1.5 = 3_. minutes. Bus arrives A Bus leaves C min min Next bus arrives B Now add to your model. The event of waiting less than 3 minutes to board a bus occurs if you arrive any time in the 1.5 minutes that a bus is sitting at the station or if you arrive during the last 3 minutes before the next bus arrives. Label a point D on the model so that it is 3 minutes before B. Then the probability is as follows: + P = _ = _ = _ 20 If you go to the station 20 times, you will probably wait less than 3 minutes to board a bus about 9_ (20) = times. Reflect 3. Discussion In the situation described in Part A, what are two ways you can find the probability that the light will not be red when you arrive? Module Lesson 2

4 Your Turn A radio station gives a weather report every 21 minutes. Each report lasts 1 minute. Suppose you turn on the radio at a random time. 4. Find the probability that the weather report will be on when you turn on the radio. 5. If you turn on the radio at 50 random times, predict how many times you will have to wait less than 1 minute before the start of the next weather report. Explain 2 Using Area to Find Probability When using area to find probability, the sample space, event, and probability are a set of points. Example 2 the trapezoid A carnival game consists of throwing a dart at the rectangular board and receiving points for hitting certain shapes. Find the probability that a dart that hits the board lands in the given shape. Round to the nearest hundredth. The area of the trapezoid is: A = 1_ = 2 ( b 1 + b 2 ) h 1_ 2 (3 + 12) (10) = 75 m 2 The area of the rectangle (board) is: 45 m 12 m 6 m 10 m 3 m 20 m A = bh = 45 (20) = 900 m 2 B The probability is: P = _ the circle The area of the circle is: A = = π ( ) 2 = ( ) π ( ) m 2 The area of the rectangle (board) is: A = ( ) = 45 ( ) = ( ) m 2 The probability is: P = _ Module Lesson 2

5 Your Turn A carnival game board consists of balloons that are 3 inches in diameter and are attached to a rectangular board. A player who throws a dart at the board wins a prize if the dart pops a balloon. 50 in. 30 in. 6. Find the probability of winning if there are 40 balloons on the board. 7. How many balloons must be on the board for the probability of winning to be at least 0.25? Elaborate 8. How are geometric probabilities different from numeric probabilities? 9. Why can real-world problems involving probability and time be solved using geometric probability? 10. Essential Question Check-In How can you use geometric probability to predict outcomes of real-world situations? Module Lesson 2

6 Evaluate: Homework and Practice 1. Arcs are drawn from the vertices of a square with a radius that is half the length of the sides of the square to produce the figure shown. If a point was chosen randomly on the figure, what is the probability of choosing a point in the middle figure? Online Homework Hints and Help Extra Practice 2. Does the probability change if the figure is duplicated multiple times? Explain using the figure shown. A bus comes to a station once every 10 minutes and waits at the station for 1.5 minutes. Assume that you arrive at the station at a random time. Express each probability as a decimal. 3. Find the probability that the bus will be at the station when you arrive. 4. Find the probability that you will have to wait more than 6 minutes to board a bus. Module Lesson 2

7 5. If you go to the station 20 times, predict about how many times you will have to wait less than 3 minutes. A radio station gives a weather report every 15 minutes. Each report lasts 45 seconds. Suppose you turn on the radio at a random time. Express each probability as a decimal. 6. Find the probability that the weather report will be on when you turn on the radio. 7. Find the probability that you will have to wait more than 5 minutes to hear any of the weather report. Round your answer to the nearest hundredth. 8. If you turn on the radio at 50 random times, predict about how many times you will have to wait less than 1 minute before the start of the next weather report. Module Lesson 2

8 A point is chosen randomly inside the large rectangle. Express the probability of each event as a decimal. Round to the nearest hundredth. 10 ft 10 ft 12 ft 6 ft 10 ft 10 ft 24 ft 18 ft 48 ft 9. The point is in the triangle. 10. The point is in the trapezoid. 11. The point is in the square. 12. The point is in the part of the rectangle that does not include the square, triangle, or trapezoid. 13. A skydiver jumps from an airplane and parachutes down to the 70-by-100-meter rectangular field shown. Assume that the skydiver will land somewhere in the field and is equally likely to land at any point within the field. Match each section of the field with the probability of landing there. A. triangle B. circle C. square D. miss all targets m 20 m 20 m 25 m 25 m Image Credits: Oliver Furrer/Cultura RM/Alamy Module Lesson 2

9 A fly lands randomly on the tangram. Express the probability of each event as a decimal. 14. The fly lands on the parallelogram. 15. The fly lands on the triangle in the upper right hand corner. 16. The fly lands on the triangle on the bottom. 17. Communicate Mathematical Ideas Would the probabilities change if you arranged the tangram pieces to form a square in a different way? Explain. In Exercises 18 21, a point is chosen at random inside the rectangle. Express the probability of each event as a decimal. Round to the nearest hundredth, if necessary. (Hint: You may want to use trigonometry to find the length of the base of the rectangle.) 18. The point is in the equilateral triangle. 30 m 10 m Module Lesson 2

10 19. The point is in the square. 30 m 10 m 20. The point is in the part of the circle that does not include the square. 21. Justify Reasoning The point is in the part of the rectangle that does not include the square, circle, or triangle. Explain. Algebra A point is chosen at random in the square region of the coordinate plane such that -5 x 5 and -5 y 5. Find the probability of each event. Round to the nearest hundredth, if necessary. 5 y 22. The point is inside the parallelogram. x The point is inside the circle The point is inside the triangle or the circle. 25. The point is not inside the triangle, the parallelogram or the circle. Module Lesson 2

11 H.O.T. Focus on Higher Order Thinking 26. Represent Real-World Problems The point value of each region of an Olympic archery target is shown in the diagram. The outer diameter of each ring is 12.2 cm greater than the inner diameter. An archer is going to shoot an arrow at the target. Assume that the arrow is going to hit the target and that the arrow is equally likely to land at any point inside the target. Find the probability of each event cm a. scoring 10 points cm b. scoring 5 or 6 points c. scoring more than 5 points Image Credits: Corbis 27. Justify Reasoning You are designing a target that is a square inside an 18 in. by 24 in. rectangle. Assume that a player will hit the target is equally likely to hit any point within the target. What size should the square be in order for the target to have a probability of 1? to have a probability of 3? Explain Analyze Relationships If a rectangle is divided into 8 congruent regions and 4 of them are shaded, what is the probability that you will randomly pick a point in the shaded area? Does it matter which of the four regions are shaded? Explain. Module Lesson 2

12 Lesson Performance Task Each spring, the Porcupine caribou herd of northern Alaska migrates some 400 miles from its winter quarters in the Canadian province of Yukon to the north coast of Alaska. To gauge the size of the herd, wildlife biologists conduct aerial photographic surveys. 1. After the herd has been thoroughly photographed, the photos are laid out in a grid like the one shown here. Assuming that individual caribou were evenly distributed across the grid, a survey conducted in July 2010 would have shown 939 caribou in grid square G9. What was the total herd population according to the survey? Explain how you found the answer A B C D E F G H I J K L M N O 2. A number of caribou have been fitted with radio collars so they can be tracked throughout the year. Find each probability. a. the probability that Caribou 16 was photographed in grid square B5 b. the probability that Caribou 5 was NOT photographed in grid square B5 c. the probability that Caribou 12 was photographed in grid square K10 and Caribou 8 was photographed in grid square D3 d. the probability that Caribou 2 was photographed in Row 11 e. the probability that Caribou 17 was photographed in Column C, D, or E 3. Suppose that in a later year, a rectangular survey grid had 20 rows, each grid square represented 344 caribou, and the probability that Caribou 7 was photographed in grid square A1 was a. How many columns would the grid square have? b. Find the caribou population. Module Lesson 2