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 ?R?1, ?R?2, and ?R?3 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?

Answer Step 1 of 6 G = {rock is granite} P(G) = 0.25 B = {rock is basalt} P(B) = 0.75 Where R1, R2, and R3 denote measured spectrum intensities at three different wavelengths R123 = {R1 < R2 < R3} R132 = {R1 < R3 < R2} R312 = {R3 < R1 < R2} P(R123 | G) = 0.60 P(R123 | B) = 0.10 P(R132 | G) = 0.25 P(R132 | B) = 0.20 P(R312 | G) = 0.15 P(R312 | G) =0.7 Step 2 of 6 a) The probability that the rock reflecting the light is granite given the wavelength reflected. P(G/R123)= P(G R123)= P(G) P(R123 | G) R123 R123 = P(G) P(R123 | G) P(G)P(R123 | G)+P(B)(R123 | B) = 0.25(0.6) 0.25(0.6)+0.75(0.1) =0.15/0.625=0.667 Step 3 of 6 the probability that the rock reflecting the light is basalt given the wavelength reflected. P(B/R123)= P(B R123)= P(B) P(R123 | B) R123 R123 = 0.75(0.1) 0.625 =0.33 P(granite R1 < R2 < R3) > P(basalt R1 < R2 < R3) I would choose granite because P(G|R123) > P(B|R123).