One disease affects one person out of a thousand. We have a test that detects it with 99% reliability, but gives false alarms in 5% of cases. My test is positive, am I at risk?

At first sight, you might think that a test with 99% reliability and a 5% false alarm rate is a fairly reliable test, and that the probability of being ill is high. Not so!

What we’re trying to estimate here is the probability of being ill knowing that a test has declared us positive. This probability, known as P(Illness | Positive), can in fact be obtained using Bayes’ Theorem, one of the main theorems of probability theory, which enables us to determine the probability of an event occurring based on another event that has already taken place, especially when these two events are interdependent:

P(Illness | Positive) = (P(Positive | Illness) x P(Illness)) / P(Positive)

Which, according to the probabilities in the statement, gives :
P(Disease | Positive) = ( 0.99 x 0.001 ) / ( 0.99 x 0.001 + 0.05 x 0.999) = 0.0194 We therefore conclude that there’s a 1.94% chance of actually being ill when you receive a positive test result. This is a very counter-intuitive result, but it makes perfect sense when approached with a good knowledge of statistics and probability!