A test for a disease correctly detects the disease in 90% of the patients that actually have the disease. Also, if a person does not have the disease the test correctly reports no disease with probability .9. Suppose 1% of the population has the disease.

(a)State carefully Bayes' Theorem for a given partition.

(b) What is the probability that the test detects the disease in a random patient?

(c) If the test detects the disease in a random patient, what is the probability that the patient has the disease?