Approximately 70% of a state sits on a shale formation from which natural gas ma

Question

Approximately 70% of a state sits on a shale formation from which natural gas may be extracted. If a geological test is positive, it has an 80% accuracy rate in correctly identifying a productive drilling site (shale is under the ground). If there is no shale under the ground, geological testing is falsely positive with a probability of 20%. Suppose the result of the geological test comes back positive (the test says there is shale under the ground). It is most important for us to know the probability that there is indeed shale under the ground given a positive geological test result. Find this probability and briefly discuss

Explanation / Answer

We need to calculate the probability that there is shale under the ground, given that test result is positive, that is,

P(shale is there | positive test result) = P(shale is there and positive test result) / P(positive test result)

= 0.7x0.8/(0.7x0.8 + 0.3x0.2)

= 0.9032

So, if the test result is positive, there is a probability of 0.9032 that there will be shale under the ground.

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