A certain test to compare the chemical similarity of gun shot residue has a true

Question

A certain test to compare the chemical similarity of gun shot residue has a true positive rate of 60% and a false positive rate of 15%. There are only a few companies that manufacture gun powder and the manufacturing methods produce a fairly consistent product. Based on this information an analyst estimates that any two random samples of gunpowder would have a 30% probability of being chemically indistinguishable.
a. What is the probability of the test resulting in a positive similarity?

b. Say a test is run on two samples an the result is positive that they have the same chemical composition. What is the probability that they are truly the same?

Explanation / Answer

a. P(the test resulting in a positive similarity) = P(Truely same)*P(true positive) + P(Truely false)*P(False Positive)

= 0.30*0.60 + 0.70*0.15 = 0.285

b. P(They are truely same give that the test result is positive)= P(True positive)/P(the test resulting in a positive similarity)

= 0.60*0.30/0.285 = 0.6315

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