Suppose an illness that can affect 1% of the people. Also assume that there is a test for that illness, that gives the correct result 99% of the times.

If you take that test, and receive a POSITIVE result, should you worry much?

If you take it again, and once more get a POSITIVE, should you worry then?

How many consecutive POSITIVEs would you have to get in order to be sure that the chances of a wrong diagnostic are 1 in a million?

(In reply to solution by Charlie)

Ok, let me see if I can explain this clearly.  The problem with the previous solution(s) is that they are working from the wrong point of information.  If a test tells me I am positive, and it will be correct 99/100, then I am positive 99/100, not 50/50.

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The confusing thing is (using the 10,000 person sample) , out of the 198 people testing positive, 99 were false positives.  If half the positives were wrong, then the odds of it being a false positive is 50%.  Right?  Actually, no.  Here’s the thing, those 198 positives were not drawn randomly, they were drawn from 2 distinct groups: those with the disease and those without.  When I make the transition from an unknown to being tested I am being “pulled” from one of those groups, right?  Now that I know I’ve been tested positive, what are the odds that I was clean and got a false result and how likely is it that I have the disease and the test simply confirmed that?  Well, 1% and 99%, respectively.

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But, how are there as many false positives and true positives?  Here’s the thing, the false positives are being drawn from a much larger sample size.  In this example there are 9,900 people who all have a chance to receive a false positive; however, there are only 100 people who have the opportunity to receive a true positive.  Given that size disparity (99-1) you end up with an equal number of results.  Of the positively tested individuals, 99% of them came from the group possessing the disease; whereas, 1% came from the non diseased group.  So, having been tested positive, there is a 1/100 chance that I came from the ‘clean but misdiagnosed’ group and a 99/100 chance that I am coming from the ‘diseased with a correct diagnosis’ group.  And, yes, I think that being 99% sure that I am sick would be cause for concern.

Posted by A on 2005-03-06 06:30:10