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 Combined probabilities by François)

You are correct.  Thanks for pointing that out.

My mistake was to misuse the sense in which the trials were independent.  Carrying on the independence assumption from an earlier post meant that each test was independent of earlier tests in the sense that there was nothing inherent in a particular person's blood, etc. that made them prone to an inaccurate result.  However, each iteration of testing is not statistically independent, because P(Test 2 Positive|Test 1 Positive) is not equal to P(Test 2 positive).

Another way to get your (correct) results is to continue to use Bayes' Theorem.

A1 = Test 1 positive  A2 = Test 2 positive

B1 = Ill                    B2 = Not Ill

P(B2|A1 and A2) = P(A1 and A2|B2)P(B2)/(P(A1 and A2|B2)P(B2)+P(A1 and A2|B1)P(B1)) = .01(.01)(.99)/(.01(.01)(.99)+.99(.99)(.01)) = 0.01, etc.

In general, P(not ill | n consecutive positive results) = 

.01^n*.99 / ((.01^n*.99)+(.99^n*.01))

Posted by Kyle on 2005-01-07 15:37:44