Question

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 measures of spectrum intensities at three different wave lengths; 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 of granite. Flights over regions of know composition have yielded the following information: Reading R_1 P(basalt|R1 < R2 < R3). If measurements yielded R1 < R2 < R3, would you classify the rock as granite? b. If measurements yielded R1 < R3 < R2, how would you classify the rock? Answer the same question forR3

Answer 1

Step-by-step explanation:

so from the question i have here, i will be giving a step by step analysis of the question.

(a). Here we are showing a relationship, i.e.

P(Granite ║ R₁ ∠ R₂ ∠ R₃) ˃ P (Basalt ║ R₁ ∠ R₂ ∠ R₃)

from the LHS;

P(Granite ║ R₁ ∠ R₂ ∠ R₃) = P(Granite ║ R₁ ∠ R₂ ∠ R₃) / P(R₁ ∠ R₂ ∠ R₃)

= P(Granite)P(R₁ ∠ R₂ ∠ R₃ ║ Granite) / [ P(Granite R₁ ∠ R₂ ∠ R₃) + P(Basaltic R₁ ∠ R₂ ∠ R₃) ]

= 0.25 × 0.6 / [(0.25×0.6)+(0.75×0.1)] = 0.667

from the RHS;

P (Basalt ║ R₁ ∠ R₂ ∠ R₃) = P(Basalt R₁ ∠ R₂ ∠ R₃) / P(R₁ ∠ R₂ ∠ R₃)

= P(Basalt)P(R₁ ∠ R₂ ∠ R₃ ║ Basalt) / [ P(Basalt R₁ ∠ R₂ ∠ R₃) + P(Granite R₁ ∠ R₂ ∠ R₃) ]

= 0.75 × 0.1 / [(0.25 × 0.6)+(0.75 × 0.1)] = 0.333

Therefore from this we can infer that;

P(Granite ║ R₁ ∠ R₂ ∠ R₃) ˃ P (Basalt ║ R₁ ∠ R₂ ∠ R₃)

(b). here we are asked to classify the rocks considering the measurement yield.

Measurement yielded R1 < R3 < R2

P(Granite ║ R₁ ∠ R₃ ∠ R₂) = P(Granite R₁ ∠ R₃ ∠ R₂) / P(R₁ ∠ R₃ ∠ R₂)

= P(Granite)P(R₁ ∠ R₃ ∠ R₂ ║ Granite) / [ P(Granite R₁ ∠ R₃ ∠ R₂) + P(Basaltic R₁ ∠ R₃ ∠ R₂) ]

= 0.25 × 0.25 / [(0.25 × 0.25)+(0.75 × 0.2)] = 0.294

also for RHS;

P (Basalt ║ R₁ ∠ R₃ ∠ R₂) = P(Basalt ║ ∠ R₃ ∠ R₂) / P(R₁ ∠ R₃ ∠ R₂)

= P(Basalt)P(R₁ ∠ R₃ ∠ R₂ ║ Basalt) / [ P(Basalt R₁ ∠ R₃∠ R₂) + P(Granite R₁ ∠ R₃ ∠ R₂) ]

= 0.75 × 0.2 / [(0.25 × 0.25)+(0.75 × 0.2)] = 0.706

from this we can infer that;

P(Granite ║ R₁ ∠ R₃ ∠ R₂) ∠ P (Basalt ║ R₁ ∠ R₃ ∠ R₂)

Also considering measurements yielded R₃ ∠ R₁ ∠ R₂

P(Granite ║ R₃ ∠ R₁ ∠ R₂) = P(Granite R₃ ∠ R₁ ∠ R₂) / P(R₃ ∠ R₁ ∠ R₂)

= P(Granite)P(R₃ ∠ R₁ ∠ R₂ ║ Granite) / [ P(Granite R₃ ∠ R₁ ∠ R₂) + P(Basaltic R₃ ∠ R₁ ∠ R₂) ]

= 0.25 × 0.15 / [(0.25×0.15)+(0.75×0.7)] = 0.067

from the RHS;

P (Basalt ║ R₃ ∠ R₁ ∠ R₂) = P(Basalt R₃ ∠ R₁ ∠ R₂) / P(R₃ ∠ R₁ ∠ R₂)

= P(Basalt)P(R₃ ∠ R₁ ∠ R₂ ║ Basalt) / [ P(Basalt R₃ ∠ R₁ ∠ R₂) + P(Granite R₃ ∠ R₁ ∠ R₂) ]

1 - 0.067 = 0.933

from this we can infer that;

P(Granite ║ R₃ ∠ R₁ ∠ R₂) ∠ P (Basalt ║ R₃ ∠ R₁ ∠ R₂)

cheers i hope this helps