It is known that in Math Town 15% of the adult population smokes. It is also known that 4 out of 10 adult smokers have the dreaded disease X, while only 5 out of 200 adult nonsmokers have this disease. A test has been developed for the detection of this disease. If the person has the disease, it will give a positive result 98% of the time. Unfortunately, 5% of the time it gives a "false positive", that is, it indicates the presence of the disease even though the patient does not have the disease.
a.) a random patient is tested and you are informed that the patient tests positive. What is the probability that the patient actually has the disease?
b.) What if you are informed that the patient is also a nonsmoker, then what is the probability that the patient actually has the disease?
c.) What if you are informed the patient who tested positive is a smoker, then what is the probability that the patient has the disease?