A test for detection of a particular disease is not fool proof. The test will correctly detect the disease 90% of the time, but will incorrectly detect the disease 1% of the time. For a large population of which an estimated 0.2% have the disease, a person is selected at random, given the test, and told that he has the disease. What are the chances that the person actually have the disease?

Question

Philoid · Accepted Answer

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

U₁ = Person actually has a disease

U₂ = Person doesn’t has a disease

A = detection of disease

From the problem

⇒

⇒ P(A|U₁) = P(Test correctly detected)

⇒

⇒ P(A|U₂) = P(Test incorrectly detected)

⇒

Now we find

P(U₁|A) = P(The person actually has the disease and correctly detected)

Using Baye’s theorem:

⇒

∴ The required probability is .