Answer:
the probability that there is indeed shale under the ground given a positive geological test result, that is, P(S|T) = 0.9032
Explanation:
Complete 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.
Solution
Let the Probability that there is shale in a randomly picked point in the state = P(S) = 70% = 0.70
The probability of the absence of shale = P(S') = 1 - 0.70 = 0.30
Let the probability of a positive test be P(T)
If a geological test is positive, it has an 80% accuracy rate in correctly identifying a productive drilling site (shale is under the ground).
That is, the probability of a positive test, given that there is shale = P(T|S) = 80% = 0.80
If there is no shale under the ground, geological testing is falsely positive with a probability of 20%.
That is, the probability of a positive test, given that there is no shale = P(T|S') = 20% = 0.20
We require the probability that there is indeed shale under the ground given a positive geological test result, that is, P(S|T)
The conditional probability P(A|B) is given mathematically as
P(A|B) = P(A n B) ÷ P(B)
Hence, the required probability,
P(S|T) = P(S n T) ÷ P(T)
But we do not have P(S n T) or P(T)
What we do have is
P(S) = 0.70
P(S') = 0.30
P(T|S) = 0.80
P(T|S') = 0.20
So, we can obtain P(S n T) and P(T) from these given probabilities.
P(T|S) = P(T n S) ÷ P(S)
P(T n S) = P(S n T) = P(T|S) × P(S) = 0.80 × 0.70 = 0.56
P(T|S') = P(T n S') ÷ P(S')
P(T n S') = P(S' n T) = P(T|S') × P(S') = 0.20 × 0.30 = 0.06
P(T) = P(S n T) + P(S' n T) = 0.56 + 0.06 = 0.62
So, we have P(S n T) and P(T) now,
P(S|T) = P(S n T) ÷ P(T)
P(S|T) = (0.56/0.62) = 0.9032258065 = 0.9032
Hope this Helps!!!