A laboratory blood test is effective in detecting a certain disease when it is in fact, present. However, the test also yields a false-positive result for of the healthy person tested (i.e. if a healthy person is tested, then, with probability , the test will imply he has the disease). If 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?
A laboratory blood test is effective in detecting a certain disease when it is in fact, present. However, the test also yields a false-positive result for of the healthy person tested (i.e. if a healthy person is tested, then, with probability , the test will imply he has the disease). If 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?

Solution:

Let represent a person who has a disease, represent a person who does not have a disease, and represent a person who has a positive blood test.

and are events that are complementary to one another.

Then

Then and

Also result is positive given that person has disease)

And (result is positive given that person has no disease)

Given a positive test result, the probability that a person has a disease is .

We have used Bayes’ theorem to arrive at our conclusion.

We acquire the values by substituting them.