Mining for dolomite.Dolomite is a valuable mineral thatis found in sedimentary rock. During mining operations, dolomite is often confused with shale. The radioactivity features of rock can aid miners in distinguishing between dolomite and shale rock zones. For example, if the gamma ray reading of a rock zone exceeds 60 API units, the area is considered to be mostly shale (and is not mined); if the gamma ray reading of a rock zone is less than 60 API units, the area is considered to be abundant in dolomite(and is mined). Data on 771 core samples in a rock quarry collected by the Kansas Geological Survey revealed the following: 476 of the samples are dolomite and 295 of the samples are shale. Of the 476 dolomite core samples, 34 had a gamma ray reading greater than 60. Of the 295 shale core samples, 280 had a gamma ray reading greater than 60. Suppose you obtain a gamma ray reading greater than 60 at a certain depth of the rock quarry. Should this area be mined?

Step by step solution

01

Important formula

The formula for probability is$\mathbf{P}\mathbf{=}\frac{\mathbf{favourableoutcomes}}{\mathbf{totaloutcomes}}$

The Baye’s formula is

$\begin{array}{rcl}\mathbf{P}\left(\frac{{\mathbf{B}}_{\mathbf{i}}}{\mathbf{A}}\right)& \mathbf{=}& \frac{\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{i}}\cap \mathbf{A}\mathbf{)}}{\mathbf{P}\mathbf{(}\mathbf{A})}\\ & \mathbf{=}& \frac{\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{i}}\mathbf{)}\mathbf{P}\left(\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{i}}}\right)}{\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{1}}\mathbf{)}\mathbf{P}\left(\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{1}}}\right)\mathbf{+}\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{2}}\mathbf{)}\mathbf{P}\left(\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{2}}}\right)\mathbf{+}...\mathbf{+}\mathbf{P}\mathbf{(}{\mathbf{B}}_{\mathbf{k}}\mathbf{)}\mathbf{P}\left(\frac{\mathbf{A}}{{\mathbf{B}}_{\mathbf{k}}}\right)}\\ & & \end{array}$

02

Find the probabilities of different samples

The probability for dolomite,Shale, gamma-ray of dolomite, gamma-ray for shale respectively.

$\begin{array}{rcl}P\left(\text{Dolomite}\right)& =& \frac{476}{771}\\ & =& 0.62\end{array}$

$\begin{array}{rcl}P\left(\text{Shale}\right)& =& \frac{295}{771}\\ & =& 0.38\\ & & \end{array}$

$\begin{array}{rcl}P\left(G|D\right)& =& \frac{34}{476}\\ & =& 0.071\\ & & \end{array}$

$\begin{array}{rcl}P\left(G|S\right)& =& \frac{280}{295}\\ & =& 0.95\\ & & \end{array}$

03

Determine if the mine area is safe to operate or not

$\begin{array}{rcl}P\left(D|G\right)& =& \frac{P\left(G\right)P\left(\frac{G}{D}\right)}{P\left(D\right)P\left(\frac{G}{D}\right)+P\left(S\right)P\left(\frac{G}{S}\right)}\\ & =& \frac{\left(0.71\right)\left(0.62\right)}{\left(0.71\right)\left(0.62\right)+\left(0.95\right)\left(0.38\right)}\\ & =& 0.1073\\ & & \end{array}$

The mine area is not safe to operate because gamma-ray reading is higher than 60.

Therefore, the probabilities are 0.62, 0.38, 0.071,and 0.95. And the mine area is not safe for operate.

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