One method of distinguishing 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, 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 yeilded the following information.

granite basalt

r1 < r2 < r3 60% 10%

r1 < r3 < r2 25% 20%

r3 < r1 < r2 15% 70%

total 100% 100%

suppose that for a randomly selected rock in a certain region, p(granite) = 0.25 and p(basalt) = 0.75.

a) show that p(granite | r1 p(basalt | r1
b) if measurements yielded r1
c) if measurements yielded r1
d) if measurements yielded r3
e) using the classification rules indicated in questions 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 or g, and both p(b) and p(g) are known].