A laboratory blood test is 99% effective in detecting a certain disease when its infection is present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1% of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

Let us assume U1, U2, U3 and A be the events as follows:


U1 = Person has disease


U2 = Person doesn’t has disease


A = Blood test result is positive


From the problem,


⇒


⇒


⇒ P(A|U1) = P(Blood test results shows positive for the person with disease)


⇒


⇒ P(A|U2) = P(Blood test results shows positive for the person without disease)


⇒


Now we find


P(U1|A) = P(The Person has a disease given that the blood test results shows positive)


Using Baye’s theorem:


⇒


⇒


⇒


⇒


⇒


⇒


∴ The required probability is .


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