Due to the positive comments I have received from my last two articles on mathematical and statistical paradoxes, I have written another one. However where the first two were interesting mathematical puzzles with minimal real world applications, this one is nastier because many people have suffered from its results. It is commonly known as the Rare Test Paradox

Suppose that you have an excellent medical test for some rare disease. Just for the sake of argument, suppose that this test is so accurate that if a patient has the disease, the test will give a positive result 99.99% of the time. And if the patient does not have the disease, the test will give a negative result 99.99% of the time. Clearly this is a very accurate test! (In the real world, most procedures are far less accurate, which makes these results scarier)

Now suppose that you test everyone in a city of one million people. It is a rare disease, so only ten people in the city actually have the disease. Most likely, all ten of those people will get positive results as the probability of a false negative is tiny. However of the 999,990 people who do NOT have the disease, the test gives a false positive to 100 people. And so in the end, 110 people are told they have the disease.

Except that of the 110 positive results, 100 are false positives. Suddenly this highly accurate test is wrong for 91% of the positive results. The test is accurate, but because the disease is so rare the mistakes will naturally outnumber the correct diagnoses.

Unfortunately this beautiful little statistical oddity is rarely if ever presented in the popular media. And that unfortunately leads to many people having far more confidence in their test results than they should, and that leads to far more stress and worry than necessary. Most healthcare providers have never been taught about this oddity, let alone know how to apply it to real world testing. With many medical tests, a positive result is almost certainly wrong and yet expensive and potentially harmful treatment must be endured as a result.

And all because of a basic misunderstanding of statistics.