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Probability Questions - Wiley

I came across these two Probability Questions while practicing Wiley QBank; Correct answers were provided but no explanations were given; hence this post. 

Q1. In the city of Townsville, 2% of the population has some kind of genetic mutation. 96% of tests correctly detect the mutation. However, 9.4% of the tests erroneously conclude that a sample has genetic mutation when in fact, it does not. If a person is shown to have a positive test result, what are the chances that he/she actually has the said genetic mutation?

Correct Answer is 17.25%

Q2. As a member of the forensics team in a murder case, you found a DNA sample which is that of the murderer’s and there is a 0.2% chance that a random person’s DNA matches this sample. In your search, one of the alleged suspects’ DNA matches with the sample you have obtained. You know that there is only a .004% chance that a random person is the actual murderer but there’s a 100% chance that the man’s DNA matches that of the murderer’s given that the man is indeed the one who killed the victim. What are the chances that the man is proven guilty given that his DNA matches your sample?

Correct Answer is 2%.


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P(A | B) = P(AB) / P(B)   ——> P(mutation | test pos ) = P( mutation & test pos ) / P( test pos)

P(AB) = (.96)(.02) = .0192

P(B) = (.094)(.98) + (.96)(.02) = .11132

P(AB) / P(B) = .0192 / .11132 = .17248 ~ 17.25%

Many thanks redsauce; now that I’ve the solution, the question seems so simple!