One method used to distinguish between granitic (G) and basaltic (B) rocks is to examine...

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 graniteR₁23 , whereas for basalt R₁23. When measurements are made remotely (using aircraft), various orderings of the R_is 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 P(granite) =.25 , and P(basalt) = .75..

a. Show thatP(granite| R₁23) >P(basalt| R₁23). If measurements yielded R₁23 , would you classify the rock as granite or basalt?

b. If measurements yielded R₁23, how would you classify the rock? Answer the same question for R₁23.

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