106. 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 Ris may arise whether the rock is basalt or granite. Flights over regions of known composition have yielded the following information:

R1 , R2 , R3 R3 , R1 , R2 Granite Basalt 60% 10%

25% 20%

R3 , R1 , R2 15% 70%

R1 , R3 , R2 R1 , R2 , R3 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?