Question

A water well is to be drilled in the desert where the soil is either​ rock, clay or sand. The probability of rock ​P(R)equals=0.53. The clay probability is ​P(C)equals=0.21. The sand probability is ​P(S)equals=0.26. If the soil is​ rock, a geological test gives a positive result with​ 35% accuracy. If it is​ clay, this test gives a positive result with​ 48% accuracy. The test gives a​ 75% accuracy for sand.
Given the test is​ positive, what is the probability that the soil is​ clay, P(clay​ | positive)? Use​ Bayes' rule to find the indicated probability.

Answer 1

The tree diagram for the probability is shown below

P(Clay|Positive) is read 'Probability of Clay given the result is Positive'.

This is a case of conditional probability.

The formula for conditional probability is given as
P(Clay|Positive) = P(Clay∩Positive) ÷ P(Positive)

P(Clay∩Positive) = 0.21×0.48 = 0.1008

P(Positive) = P(Rock∩Positive) + P(Clay∩Positive) + P(Sand∩Positive)
P(Positive) = (0.53×0.53) + (0.21×0.48) + (0.26×0.75)
P(Positive) = 0.2809 + 0.1008 + 0.195
P(Positive) = 0.5767

Hence,
P(Clay|Positive) = 0.1008÷0.5767 = 0.175 (rounded to 3 decimal place)