There is a method to distinguish granite (G) or basaltic (B) rocks in the soil, the method consists of examining a portion of the soil by means of the infrared spectrum that the rocks reflect on the soil surface. Let R1, R2 and R3 be intensity levels at different wavelengths of the infrared spectrum. Generally for granite the intensities are presented in the order R1 < R2 < R3, while for basalt they are presented in the order R3 < R1 < R2. When measurements are made remotely (aerially), different orders of the Ris can be presented, whether they are basalt or granite rocks. Several measurements on a region whose soil composition is known have yielded the following information.
There is a method to distinguish granite (G) or basaltic (B) rocks in the soil, the method consists of examining a portion of the soil by means of the infrared spectrum that the rocks reflect on the soil surface. Let R1, R2 and R3 be intensity levels at different wavelengths of the infrared spectrum. Generally for granite the intensities are presented in the order R1 < R2 < R3, while for basalt they are presented in the order R3 < R1 < R2. When measurements are made remotely (aerially), different orders of the Ris can be presented, whether they are basalt or granite rocks. Several measurements on a region whose soil composition is known have yielded the following information.
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Transcribed Image Text:There is a method to distinguish granite (G) or basaltic (B) rocks in the soil, the
method consists of examining a portion of the soil by means of the infrared
spectrum that the rocks reflect on the soil surface. Let R1, R2 and R3 be intensity
levels at different wavelengths of the infrared spectrum. Generally for granite the
intensities are presented in the order R1 < R2 < R3, while for basalt they are
presented in the order R3 < R₁1 < R2. When measurements are made remotely
(aerially), different orders of the Ris can be presented, whether they are basalt or
granite rocks. Several measurements on a region whose soil composition is known
have yielded the following information.
Order
Granite
R₁ R₂ R3 60%
R₁ < R3 < R₂
25%
R3 < R₁ < R₂
15%
Basalt
10%
20%
70%
Assuming that if a rock is randomly selected in a certain region P(Granite) = 0.25 and P(Basalt)
= 0.75.
a) Show that P(Granite|R1 < R2 < R3) > P(Basalt|R1 < R2 < R3).
b) If new measurements resulted in the order of wavelengths R1 < R2 < R3, how would you
classify the rock, granite or basalt?
c) If they were in the order R3 < R₁1 < R2, how would you classify them?
Expert Solution
Step 1
The event G denotes that the rock is Granite and the event B denotes that the rock is Basaltic. It is given that
Also, there are three order of intensities given. The likelihood probabilities are:
| Order | Granite | Basalt |
| R1<R2<R3 | 0.6 | 0.1 |
| R1<R3<R2 | 0.25 | 0.2 |
| R3<R1<R2 | 0.15 | 0.7 |
In this table, the probabilities of the three order of intensities given that the rock was G or B are shown.
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