Awater 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.

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)