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EXAMPLE 1

Find the probability that a point chosen at random on PQ is on RS. –. –. 6. 3. Length of RS Length of PQ. 4 ( 2) 5 ( 5). ,. =. =. P ( Point is on RS ) =. =. –. –. 10. 5. EXAMPLE 1. Use lengths to find a geometric probability. SOLUTION. 0.6 , or 60%.

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EXAMPLE 1

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  1. Find the probability that a point chosen at random on PQis on RS. – – 6 3 Length of RS Length of PQ 4 ( 2) 5 ( 5) , = = P(Point is on RS)= = – – 10 5 EXAMPLE 1 Use lengths to find a geometric probability SOLUTION 0.6, or 60%.

  2. EXAMPLE 2 Use a segment to model a real-world probability MONORAIL A monorail runs every 12 minutes. The ride from the station near your home to the station near your work takes 9 minutes. One morning, you arrive at the station near your home at 8:46. You want to get to the station near your work by 8:58. What is the probability you will get there by 8:58?

  3. EXAMPLE 2 Use a segment to model a real-world probability SOLUTION STEP 1 Find: the longest you can wait for the monorail and still get to the station near your work by 8:58. The ride takes 9 minutes, so you need to catch the monorail no later than 9 minutes before 8:58, or by 8:49. The longest you can wait is 3 minutes (8:49 – 8:46 = 3 min).

  4. Model the situation. The monorail runs every 12 minutes, so it will arrive in 12 minutes or less. You need it to arrive within 3 minutes. EXAMPLE 2 Use a segment to model a real-world probability STEP 2 The monorail needs to arrive within the first 3 minutes.

  5. P(you get to the station by 8:58) Favorable waiting time 3 1 = = = Maximum waiting time 12 4 The probability that you will get to the station by 8:58. is 1 ANSWER or 25%. 4 EXAMPLE 2 Use a segment to model a real-world probability STEP 3 Find: the probability.

  6. Find the probability that a point chosen at random on PQis on the given segment. Express your answer as a fraction, a decimal, and a percent. RT 1. 1 ANSWER , 0.1, 10% 10 for Examples 1 and 2 GUIDED PRACTICE

  7. TS Find the probability that a point chosen at random on PQis on the given segment. Express your answer as a fraction, a decimal, and a percent. 2. ANSWER 1 , 0.5, 50% 2 for Examples 1 and 2 GUIDED PRACTICE

  8. Find the probability that a point chosen at random on PQis on the given segment. Express your answer as a fraction, a decimal, and a percent. PT 3. ANSWER 2 , 0.4, 40% 5 for Examples 1 and 2 GUIDED PRACTICE

  9. Find the probability that a point chosen at random on PQis on the given segment. Express your answer as a fraction, a decimal, and a percent. RQ 4. ANSWER 7 , 0.7, 70% 10 for Examples 1 and 2 GUIDED PRACTICE

  10. ANSWER 1 or 50%. 2 for Examples 1 and 2 GUIDED PRACTICE 5. WHAT IF?In Example 2, suppose you arrive at the station near your home at 8:43. What is the probability that you will get to the station near your work by 8:58?

  11. ARCHERY EXAMPLE 3 Use areas to find a geometric probability The diameter of the target shown at the right is 80 centimeters. The diameter of the red circle on the target is 16 centimeters. An arrow is shot and hits the target. If the arrow is equally likely to land on any point on the target, what is the probability that it lands in the red circle?

  12. Area of red circle P(arrow lands in red region) = Area of target (82) 64 1 = = = (402) 1600 25 ANSWER The probability that the arrow lands in the red region is 1 , or 4%. 25 EXAMPLE 3 Use areas to find a geometric probability SOLUTION Find the ratio of the area of the red circle to the area of the target.

  13. SCALE DRAWING Find the area of the field. The shape is a rectangle, so the area is bh = 10 3 = 30 square units. EXAMPLE 4 Estimate area on a grid to find a probability Your dog dropped a ball in a park. A scale drawing of the park is shown. If the ball is equally likely to be anywhere in the park, estimate the probability that it is in the field. SOLUTION STEP 1

  14. Find the total area of the park. EXAMPLE 4 Estimate area on a grid to find a probability STEP 2 Count the squares that are fully covered. There are 30 squares in the field and 22 in the woods. So, there are 52 full squares. Make groups of partially covered squares so the combined area of each group is about 1 square unit. The total area of the partial squares is about 6 or 7 square units. So, use 52 + 6.5 =58.5 square units for the total area.

  15. Area of field 300 30 20 P(ball in field) = = = Total area of park 58.5 39 585 The probability that the ball is in the field is about 20 , or 51.3%. ANSWER 39 EXAMPLE 4 Estimate area on a grid to find a probability STEP 3 Write a ratio of the areas to find the probability.

  16. 14 ANSWER 25 = 56% for Examples 3 and 4 GUIDED PRACTICE 6. In the target in Example 3, each ring is 8 centimeters wide. Find the probability that an arrow lands in a black region.

  17. 19 or about 48.7%. ANSWER 39 for Examples 3 and 4 GUIDED PRACTICE 7. In Example 4, estimate the probability that the ball is in the woods.

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