Deadly Disease

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There is a rare but very serious disease which is usually fatal for those who contract it. It is known to affect about 1 person out of every 100,000.

However, a recently developed test is 99% accurate at determining whether or not an individual has it.

You take the test, and it returns a positive result to the existence of the disease.

What is the probability that you do actually have the disease?


Comments

1. nima -- 2008-09-10 19:18

Here's my crack at this... 

P(Disease|Positive) = (99/10,000,000)/((99/10,000,000)+(99999/10,000,000))

i.e. The probability that I have the disease, given that the test says so, is 0.099%.

2. gpdawson -- 2008-09-14 07:23

To express it narratively, if 10,000,000 took the test then:

100 would actually have it - and test would return 99% positive results = 99 positive results

9,999,900 would not have it - but test would return 1% positive results = 99,999 positive results

So there would be 99,999+99 = 100098 positive results per 100 people who actually had it, and the probability that positive result means you actually have it = 100 / 100098 = 0.0999021%

This is almost identical to your answer, Nima. It would be identical if the first 99 in your expression was changed to 100.

Actually when I first posed this question, I was thinking 0.1% would have been an acceptably accurate answer. Thanks for making things a bit more rigorous!