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

, whereas for basalt . 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:

Suppose that for a randomly selected rock in a certain region, and .

a. Show that . If measurements yielded , would you classify the rock as granite or basalt?

b. If measurements yielded , how would you classify the rock? Answer the same question for .

c. Using the classification rules indicated in parts

(a) and (b), when selecting a rock from this region, what is the probability of an erroneous classification? [Hint: Either G could be classified as B or B as G, and P(B) and P(G)
are known.]

d. If rather than .25, are there values of p (other than 1) for which one would always classify a rock as granite?