Question

An water well is to be drilled in the desert where the soil is either rock, clay or sand. The probability of rock P(R) = 0.53. The clay probability is P(C) = 0.21. The sand probability is P(S) = 0.26. It if it 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 soil is rock, P(sand | positive)?

Answer 1

The wording of this question confuses me. Are there 3 different tests? One for rock, clay, and sand each? If there are 3 tests, then we would need to know how often a correct positive result would occur and how often a false positive result would occur.

If it's simply one binary test, then solving it involves the ratio of correct positive rock results P(R) * rAccuracy divided by the sum of P(C) * cAccuracy + P(S) * sAccuracy + P(R) * rAccuracy.