A terrible disease sweeps around the world, luckily only affecting 1

Should be .99

I do not see any catch in this question.. I might be missing something.

If you want to see the catch, think of more extreme numbers:

Say 1 person in the world has disease ZZZ, and 10 billion people do not. There's a test that is 50% reliable and you take it. If you test "positive", is there a 50% chance that you have the disease? Surely not; billions of people will test positive who don't have ZZZ.

There is a profound difference between the following two questions:

If you have the disease, what is the probability you test positive? (the answer is 99%)

If you test positive, what is the probability you have the disease? (the answer is not 99%)

The original post asks the second question, not the first.

Great question on conditional probability.

P(A happening / given B occured) = P(A AND B)/P(B)

P(You have disease/given outcome is positive) = P(you have disease AND outcome is positive)/P(outcome is positive)

P(outcome is positive) = false positive + true positive = \((9999/10000)\) * \((1/100)\) + \((1/10000)\) * \((99/100)\)

P(you have disease AND outcome is positive) = \((1/10000) * (99/100)\)

So P(A given B) = P(You have disease/given outcome is positive) = \((1/10000) * (99/100)\) / (\((9999/10000)\) * \((1/100)\) + \((1/10000)\) * \((99/100)\))

= \(99 / (9999 + 99)\)

= 1/ (1 + \(9999/99\)) = 1 / (1 + 101) = 1/ 102

Answer (B)

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