# Accuracy and prediction… the limits of our intuition

Imagine that an infectious disease is spreading around the world, and that a test is available, with for example a proven accuracy of 90%. Ask yourself the following question:

What is the likelihood that I am infected, if the test is positive?

Usually, people answer that the probability is 90%, that is, equal to the accuracy of the test. But this answer is wrong and betrays our difficulty in reasoning correctly with probabilities.

**In reality, the probability in question could be any number between 0% and 100%!**

Now I’ll explain. Before doing so, however, a clarification is necessary. A test has two types of accuracy. One that allows it to detect infected people, which is called ‘sensitivity’, and one that allows it to detect non-infected people, which is called ‘specificity’. But to simplify the discussion, we can here consider that the 90% accuracy of the test in question means that both its sensitivity and its specificity are 90%.

So, how is it possible that with a 90% accurate test, the probability that a person is infected, when the test response is positive, can be any percentage, between 0% and 100%?

It’s simple. It is because we usually forget that this probability depends on how many infected persons are present in the…