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?

I don't agree with your formula of powers of 0.5 to generalize the probability of false positive, which finds that you need about 20 tests to conclude to one chance over 1 million.

If pn is the probability of false positive after n tests, then p(n+1) is :

p(n+1)=1/(1+99*((1-pn)/pn))

So, as p1=0.5 :

p2=1/100=0.01
p3=1.02 e-4
p4=1.03 e-6 which is still slightly larger than 1/10^6
and p5=1.04 e-8

The probability of false positive is not divided by 2 at each positive test, it is nearly divided by 100 (especially from the 2nd step where the probability starts to be very small).

Actually, after step number 4, the probability is useless in this example because the population of the world is not big enough for these results to mean anything in statistics.

Posted by François on 2005-01-06 17:16:22