A researcher develops a blood test intended to detect if a person has a genetic

Question

A researcher develops a blood test intended to detect if a person has a genetic disease; the result of the test is either "yes" or "no". Unfortunately, the test is not perfect, and the test confuses having the disease with having red hair. The test result is "yes" with probability 99% if the tested person has the genetic disease or has red hair, the test result is "no" with probability 99% if the tested person does not have genetic disease and does not have red hair The probability of a person having the genetic disease is 1%, and the probability of a person having red hair is also 1%. If having the genetic disease is independent of having red hair, what is the probability that a tested person has the genetic disease if the test returns "yes"?

Explanation / Answer

P( Y/D) = 0.99, P(D)=0.01

P(Y/ND) =P(yes/non disease)= 0.01 ; P(ND)=0.99

P(Y/R)=0.99, P(R)=0.01

P(Y/NR) = 0.01, P(NR) = 0.99

P( D/Y) = P(D)*P(Y/D) / Sum( P(D)*P(Y/D) + P(ND)*P(Y/ND)+P(R)*P(Y/R) + P(NR)*P(Y/NR) = 0.99*0.01 / (0.99*0.01+0.99*0.01+0.01*0.99+0.01*0.99) = 1/4= 0.25

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