A rare genetic disorder occurs in 2 out of 1000 people 0002

A rare genetic disorder occurs in 2 out of 1000 people (0.002 or 0.2%) which causes people to laugh at Will Ferrell movies. There is a diagnostic test for this disorder that has a sensitivity of 95% and a specificity of 96%. You take the test and your doctor tells you that you have tested positive. What is the probability that you indeed do have this disorder?

Solution

Here,

P(positive and with disorder) = P(with disorder)*sensitivity = 0.002*0.95 = 0.0019

Now,

P(positive) = P(with disorder)*sensitivity + P(without disorder)*(1-specificity) = 0.002*0.95 + (1-0.002)*(1-0.96) = 0.04182

Thus,

P(with disorder|positive) = P(positive and with disorder) / P(positive) = 0.0019/0.04182

= 0.045432807 [ANSWER]