The length of the segment between 2 and 10 is 10 – 2 = 8.

1. 2 3 P(landing between 2 and 10) = = , or length of favorable segment length of entire segment 8 12 Geometric Probability LESSON 10-8 Additional Examples A gnat lands at random on the edge of the ruler below. Find the probability that the gnat lands on a point between 2 and 10. The length of the segment between 2 and 10 is 10 – 2 = 8. The length of the ruler is 12. Quick Check

2. Represent this using a segment. 3 4 P(waiting more than 15 minutes) = , or 3 4 45 60 The probability that Benny will have to wait at least 15 minutes is , or 75%. Geometric Probability LESSON 10-8 Additional Examples Quick Check A museum offers a tour every hour. If Benny arrives at the tour site at a random time, what is the probability that he will have to wait at least 15 minutes? Because the favorable time is given in minutes, write 1 hour as 60 minutes. Benny may have to wait anywhere between 0 minutes and 60 minutes. Starting at 60 minutes, go back 15 minutes. The segment of length 45 represents Benny’s waiting more than 15 minutes.

3. Find the area of the circle. Because the square has sides of length 20 cm, the circle’s diameter is 20 cm, so its radius is 10 cm. A = r 2 = (10)2 = 100 cm2 Find the area of the region between the square and the circle. A = (400 – 100 ) cm2 Geometric Probability LESSON 10-8 Additional Examples A circle is inscribed in a square target with 20-cm sides. Find the probability that a dart landing randomly within the square does not land within the circle. Find the area of the square. A = s2 = 202 = 400 cm2

4. area between square and circle area of square P (between square and circle) = = = 1 – 0.215 400 – 100 400 4 Geometric Probability LESSON 10-8 Additional Examples (continued) Use areas to calculate the probability that a dart landing randomly in the square does not land within the circle. Use a calculator. Round to the nearest thousandth. The probability that a dart landing randomly in the square does not land within the circle is about 21.5%. Quick Check

5. To win a prize, you must toss a quarter so that it lands entirely between the two circles below. Find the probability that this happens with a quarter of radius in. Assume that the quarter is equally likely to land anywhere completely inside the large circle. 15 32 The center of a quarter with a radius of in. must land at least in. beyond the boundary of the inner circle in order to lie entirely outside the inner circle. Because the inner circle has a radius of 9 in., the quarter must land outside the circle whose radius is 9 in. + in., or 9 in. 15 32 15 32 15 32 15 32 Geometric Probability LESSON 10-8 Additional Examples

6. Similarly, the center of a quarter with a radius of in. must land at least in. within the outer circle. Because the outer circle has a radius of 12 in., the quarter must land inside the circle whose radius is 12 in. – in., or 11 in. 15 32 15 32 15 32 17 32 17 32 Find the area of the circle with a radius of 11 in. A = r2 = (11 )2 417.73672 in.2 17 32 Geometric Probability LESSON 10-8 Additional Examples (continued) 15 32 Find the area of the circle with a radius of 9 in. A = r2 = (9 )2 281.66648 in.2 15 32

7. 417.73672 – 281.66648 417.73672 P (outer region) = 0.32573 = 136.07024 417.73672 area of outer region area of large circle Geometric Probability LESSON 10-8 Additional Examples (continued) Use the area of the outer region to find the probability that the quarter lands entirely within the outer region of the circle. The probability that the quarter lands entirely within the outer region of the circle is about 0.326, or 32.6%. Quick Check