I have the following task: There is 4% of sick people in a population. There is 5% chance for a positive test for a healthy person. There is 2% chance for a negative test for a a sick person. We tested 100 people and 4 had a positive test. What is the probability of such an outcome?

So far I did this:

+ means positive test, D means diseased

D = 0.04, P(+|D) = 0.98, P(+|!D) = 0.02

P(+|D) = P(+∩D) / P(D) => P(+∩D) = P(D) * P(+|D) = 0.0392

P(+|!D) = P(+∩!D) / P(!D) => P(+∩!D) = P(!D) * P(+|!D) = 0.048

P(+) = P(+∩D) + P(+∩!D) = 0,0872

So I found that the probability for a test to be positive is 0,0872, but I am not sure what to do next.

So far I did this:

+ means positive test, D means diseased

D = 0.04, P(+|D) = 0.98, P(+|!D) = 0.02

P(+|D) = P(+∩D) / P(D) => P(+∩D) = P(D) * P(+|D) = 0.0392

P(+|!D) = P(+∩!D) / P(!D) => P(+∩!D) = P(!D) * P(+|!D) = 0.048

P(+) = P(+∩D) + P(+∩!D) = 0,0872

So I found that the probability for a test to be positive is 0,0872, but I am not sure what to do next.

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