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Problem 3. One method used to distinguish between granitic (G) and basaltic (B) rocks is to examine a portion of the infrared spectrum of the sun's energy reflected from the rock surface. Let R1, R2, and R3 denote measured spectrum intensities at three different wavelengths: Typically, for granite R₁ < R2 < R3, whereas for basalt R3

Problem 3. One method used to distinguish between granitic (G) and basaltic (B) rocks is to examine a portion of the infrared spectrum of the sun's energy reflected from the rock surface. Let R1, R2, and R3 denote measured spectrum intensities at three different wavelengths: Typically, for granite R₁ < R2 < R3, whereas for basalt R3

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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Problem 3. One method used to distinguish between granitic (G) and basaltic (B) rocks is to examine a portion of the infrared spectrum of the sun's energy reflected from the rock surface. Let R1, R2, and R3 denote measured spectrum intensities at three different wavelengths: Typically, for granite R₁ < R2 < R3, whereas for basalt R3 <R₁ <R2. When measurements are made remotely using aircraft, various orderings of the Ri's may arise whether the rock is basalt or granite. Flights over regions of known composition have yielded the following information: Granite Basalt R1 < R2 < R3 60% 10% R1 < R3 < R2 25% 20% R3 <R1 <R2 15% 70% Suppose that, for a randomly selected rock in a certain region, P(granite) = 0.25, and P(basalt) = 0.75. a. If measurements yielded R₁ < R2 < R3, would you classify the rock as granite or basalt based on conditional probabilities P(granite | R₁ <R2 < R3) and P(basalt | R1 < R2 < R3)? b. If measurements yielded R₁ < R3 < R2, how would you classify the rock based on conditional probabilities P(granite | R1 < R3 < R2) and P(basalt | R1 < R3 <R2)? C. If measurements yielded R3 < R₁ < R2, how would you classify the rock based on conditional probabilities P(granite | R3 < R1 < R2) and P(basalt | R3 <R1 < R2)? d. If P(granite) = p rather than 0.25, are there values of p (other than 1) for which one would always classify a rock as granite based on P(granite | R1 < R2 < R3), P(granite | R₁ < R3 < R2), and P(granite | R3 <R₁ <R2)?
Transcribed Image Text:Problem 3. One method used to distinguish between granitic (G) and basaltic (B) rocks is to examine a portion of the infrared spectrum of the sun's energy reflected from the rock surface. Let R1, R2, and R3 denote measured spectrum intensities at three different wavelengths: Typically, for granite R₁ < R2 < R3, whereas for basalt R3 <R₁ <R2. When measurements are made remotely using aircraft, various orderings of the Ri's may arise whether the rock is basalt or granite. Flights over regions of known composition have yielded the following information: Granite Basalt R1 < R2 < R3 60% 10% R1 < R3 < R2 25% 20% R3 <R1 <R2 15% 70% Suppose that, for a randomly selected rock in a certain region, P(granite) = 0.25, and P(basalt) = 0.75. a. If measurements yielded R₁ < R2 < R3, would you classify the rock as granite or basalt based on conditional probabilities P(granite | R₁ <R2 < R3) and P(basalt | R1 < R2 < R3)? b. If measurements yielded R₁ < R3 < R2, how would you classify the rock based on conditional probabilities P(granite | R1 < R3 < R2) and P(basalt | R1 < R3 <R2)? C. If measurements yielded R3 < R₁ < R2, how would you classify the rock based on conditional probabilities P(granite | R3 < R1 < R2) and P(basalt | R3 <R1 < R2)? d. If P(granite) = p rather than 0.25, are there values of p (other than 1) for which one would always classify a rock as granite based on P(granite | R1 < R2 < R3), P(granite | R₁ < R3 < R2), and P(granite | R3 <R₁ <R2)?
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