In this video by Oxford mathematician Peter Donnelly, he poses the following hypothetical question:

Given an AIDS test which is 99% accurate, what is the probability you have AIDS if you tested positive.

The interesting part is this isn’t as easy as it sounds, and the rarity of the disease determines the answer as much as anything else. Let’s say for the sake of discussion that on average out of 1 million people, 100 have AIDS. We would then see the following:

1. Of the 100 people who actually have AIDS, 99 people would correctly test positive. One person would have AIDS and the test wouldn’t catch it.
2. Of the remaining 999,900 people who do not have AIDS, 9,999  would incorrectly test positive.

This means that of the 1 million people who get tested for AIDS, 10,098 of them would test positive, while only 99 actually have AIDS.

You can take a test which is 99% accurate, have it turn out positive, and still only have a 99/10,098 = 0.98% chance that you actually have AIDS.  An extremely accurate test says you have AIDS, and yet it’s extremely unlikely that you do.

This leads me to agree with former Prime Minister Benjamin Disraeli, who famously stated, “There are three kinds of lies: lies, damned lies, and statistics.”