9.2.3 Geometric Probability (continued) Example 3 Using Angle Measures to Find Geometric Probability
The angle measure model is used here to demonstrate probabilities using a spinner. The measures of the central angles dividing up the circle are given. The probability of the pointer landing on the yellow area is the ratio of the central angle of the yellow area and 360°. Thus, P = 120°/360° = 1/3. The probability of the pointer landing on blue or green is the sum of the two individual probabilities. Thus, P = 55°/360° + 25°/360° = 80°/360° = 2/9. The probability of the pointer not landing on purple is equal to 1 minus the probability of it landing on purple. The central angle of the purple sector is not given, but it is equal to 360° minus 100°, which is the sum of all the other angles. Thus, P = 1 − (100°/360°) = 260°/360° = 13/18.
Example 4 Using Area to Find Geometric Probability
The area model is used to demonstrate geometric probability in this example. The measures of various dimensions of the sample space and the included figures are given. The sample space is the area of the large rectangle. The area of the sample space is 120 · 50 = 6000 m 2 . The probability that a randomly chosen point will lie inside the isosceles triangle is equal to the ratio of the area of the triangle to the area of the large rectangle. The area of the triangle can be calculated from the given base length and height. The probability is P = 600/6000 = 1/10 = 0.1. The probability that a randomly chosen point will lie inside the circle is equal to the ratio of the area of the circle to the area of the large rectangle. The area of the circle can be calculated from the given radius length. The probability is P = 100 π /6000 ≈ 0.05.
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