Question: A common test for tuberculosis (TB) is a skin test where a substance is injected into a patient's arm to see if a rash develops

A common test for tuberculosis (TB) is a skin test where a substance is injected into a patient's arm to see if a rash develops in a few days. The test is relatively accurate, but in a few cases a rash might be detected even when the patient does not have TB (a false positive). Assume that the probability of the first event P (Rash if not TB), is 5% and the chance of the other, P (not Rash if TB), is 1%. Suppose also that only 4 in 10,000 people have TB. An applicant for a teaching position is required to have a TB test and the test comes back positive (shows rash). What is the probability that the applicant really has TB?

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