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Are you ill? (Posted on 2005-01-03) Difficulty: 3 of 5
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?

See The Solution Submitted by e.g.    
Rating: 3.7500 (4 votes)

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solution | Comment 3 of 18 |

Out of 10,000 people, 100 will have the disease, 99 of whom will be correctly diagnosed (positive test result).  9,900 will not have the disease, but 99 will be erroneously diagnosed as having the disease (positive result).

So if you've gotten a positive result (i.e., a test indication that you have the disease), there is a 50% chance you actually have the disease.  How much you worry over such a probability is a matter of psychology.

The problem of successive tests depends on what manner of unreliability the 1% is.  Is it that certain types of the disease resist detection, and some people's chemistry invariably leads to false positives?  In that case, repeat testing will not change the result, or the probability.

If however the tests' error probability is independent of the iteration of the test, then a double test with the same result has only a .01^2 = .0001 probability of coming out wrong.  The probability you have the disease after two independent positive results is .01*.99*.99/(.01*.99*.99+.99*.01*.01) = .99.  That would be a 99% chance you have the disease.  I'd very much likely worry then.

After 4 positive results, the probability you have the disease would be about 0.99999896939, or just about 1 chance in a million that you don't have the disease.  Again, that's assuming the test results' error chances are independent.

  Posted by Charlie on 2005-01-03 18:01:27
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