A medical math problem -- the meaning of a false positive

Swampy goes to the doctor. A gorgeous nurse takes his temperature, weighs him, and leads him to an examination room. A tall dark haired doctor who looks exactly like Lisa Edelstein (okay, Swampy has been watched too many episodes of Dr House recently) examines him. During the examination, she takes some blood.

Then comes the moment when the doctor walks in the room, looking serious.

"I have some news for you, Swampy"

"What, you are about to divorce your husband and run away with me to a tropical island paradise where we will open a fish taco stand and live out our days together?"

"No, Swampcritter, this is serious. We have tested your blood, and the initial results for Dreaded Disease are positive."

Swampy thinks to himself, this explains it. The daily cravings for food and water, the nightly sleep sessions...

"What are Swampy's chances, doctor?"

"Well -- we will need to send your blood to a lab for a final answer. You should be aware that our screening test has a 5% false positive rate."

"Okay, Doc, what else should I know?"

"Here is a pamphlet on Dreaded Disease. It explains that it occurs in 2 out of every 1000 people, and quite a bit is known about treatments"

Okay, now having heard the story -- what is the actual chance of Swampy being sick?


Most people would look at the first fact, that of a 5% false positive rate, and say that the chance of being sick is 95%. Thus, the positive on the screening test is viewed could be quite worrying.

However, its important to also consider the other fact given, the rate of the disease in the population.

Think of it this way. Give the screening test to 1000 random people. With a 5% false positive rate, 1000 * 0.05 = 50 people will test positive. But we also know that only 2 people actually have the disease. So the odds of having the disease are 2/50, or 4%.

Whew.