What is the chance that a person found to have a positive result actually has the disease?

The correct answer: Around 2%

"The commonsense way to think mathematically about the problem is this: Only 1 person in 1000 has this disease, as compared with about 50 in 1000 who will get a false positive result (5% of 999). It is far more likely that any one person who tests positive will be one of the 50 false positives than the 1 true positive. In fact, the odds are 1 in 51 that any one person who tests positive actually has the disease, and that translates into only a 2% chance, even in light of the positive test."

"In a study at a prominent medical school, physicians, residents, and fourth-year medical students were asked [this] question...Almost half of the respondents answered 95%. Only 18% of the group got the correct answer. Those answering incorrectly were...failing to take into account the importance of the base-rate information, namely, (only) 1 person among 1000 tested will have the disease...Considering that even highly educated medical personnel can make errors in understanding probabilistic data of this kind, we should not be at all surprised that probability often seems to be at odds with the intuitive judgments of their patients and other ordinary people."

--pages 3-4, 7 of Randomness

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