## The Rare Test Paradox

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.

In : Mathematics