Problem 1) (10 points) A common test for tuberculosis (TB) is a skin test where

Question

Problem 1) (10 points) A common test for tuberculosis (TB) is a skin test where a substance is injected into a relatively accurate, but in a few cases a rast might be detected even when the subject does not have TB (a false positive) or the rash may not be seen even when the subject has TB (a false negative). Assume that the probability of the rash being detected when there is no TB s about 5% and the chance of rash not being detected when there sTBis about 1%. Suppose also that about 4 in 1000 people have TB. An applicant for a teaching position is required to get a TB test and the test comes back positive (ie, rash is detected). What is the probability that the applicant really has TB? (Round your answer to three decimal places)

Explanation / Answer

Ans:

Given that

P(positive/no TB)=0.05

P(negative/TB)=0.01

P(TB)=4/1000=0.004

P(positive)=P(positive/TB)*P(TB)+P(positive/no TB)*P(no TB)

=(1-0.01)*0.004+0.05*(1-0.004)

=0.05376

P(TB/positive)=P(positive/TB)*P(TB)/P(positive)

=(1-0.01)*0.004/0.05376

=0.074

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