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Question -

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?



Answer -

Let E1 bethe event that person has a disease, E2 be the event thatperson don not have a disease and A be the event that blood test is positive.

As E1 andE2 are the events which are complimentary to each other.

Then P (E1)+ P (E2) = 1

⇒ P (E2)= 1 – P (E1)

Then P (E1)= 0.1% = 0.1/100 = 0.001 and P (E2) = 1 – 0.001 = 0.999

Also P (A|E1)= P (result is positive given that person has disease) = 99% = 0.99

And P (A|E2)= P (result is positive given that person has no disease) = 0.5% = 0.005

Now the probabilitythat person has a disease, give that his test result is positive is P (E1|A).

By using Bayes’theorem, we have

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