1. ## smokers (conditional prob.)

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?

2. Originally Posted by ihavvaquestion
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?
These problems are common and have a basic solution method that makes a lot of sense once you see it, but might be elusive before you're shown how.

See this example on Wikipedia, and this thread, post #4. (The problem on the other thread is different in form but equivalent in content.)

By the way the smoker/non-smoker distinction makes this slightly more complex than usual, and if you need further guidance I can write something up.