Suppose there are two different diagnostic tests (say test A and test B) for some disease of interest. Assume that the prevalence of this disease in a large population is 1%. Test A has a false negative rate of 10% (false negative means that the test result is negative when the test is applied to a person who has the disease). Similarly, the false negative rate of test B is 5%. The false positive rate of test A is 4% (false positive means that the test result is positive even though it is applied to a person who does not have the disease). Similarly, the false positive rate of test B is 6%.

Define the events

A= test A is positive

B = test B is positive

The events A and B are conditionally independent. That means that if the disease status of a person is known (as either D, has disease, or as D ', does not have disease), then

P (A B|D) = P (A|D)P (B|D)

P(A B|D ') = P(A|D ')P(B|D ')

1. Given that test A is positive when administered to a person chosen at random from the population, what is the probability that test B will also be positive for the same person?
2. If both tests A and B are positive when administered to a person selected at random from the population, what is the probability this person has the disease?
3. Now suppose the tests are applied to a person with unknown disease status. Are the events A and B independent in this case? Explain why or why not.