Conditional Probability Problem

[pocketjacks]

Suppose that if a person with tb is given a chest X-ray the probability that his condition will be detected is .95, and that if a person without tb is given a chest X-ray the probability that he will be diagnosed incorrctly as having tb is .002. Suppose, furthermore, that .1 per cent of the adult residents of a certain city have tb. If one of these persons selected at random is diagnosed as having tb on the basis of a chest X-ray, what is the probability that he actually has tb?

Why is the answer not just the complement of .002 which is .998? Or is it?

Do you really have to consider the last detail of the problem?

Thank you very much in advance.