A laboratory blood test is 99% effective in detecting a certain disease when it is in fact, 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 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?

Given: let E_{1} be the event that person has a disease, E_{2} be the event that person don not have a disease and A be the event that blood test is positive.

As E_{1} and E_{2} are the events which are complimentary to each other.

Then P (E_{1}) + P (E_{2}) = 1

⇒ P (E_{2}) = 1 - P (E_{1})

Then

and P (E_{2}) = 1 – 0.001 = 0.999

Also P(A|E_{1}) = P (result is positive given that person has disease) = 99% = 0.99

And P(A|E_{2}) = P (result is positive given that person has no disease) = 0.5% = 0.005

Now the probability that person has a disease, give that his test result is positive is P(E_{1}|A).

By using bayes’ theorem, we have:

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