Statistics is Unintuitive (Part 2)

Here is another day to day situation where statistics can be very unintuitive.

Lets say we have a medical test that has 99 % accuracy. You take the test and it comes out positive. What is the chance that you are actually positive? Common sense would suggest that there is a 99 % chance you are positive, and a 1 % chance the test gave the wrong result.

Let us do the calculation step by step and see the actual answer.

First, we need one more piece of information - what is the positivity in the population. Lets assume that 100 people per lakh are actually positive. So, 99,900 people are negative per lakh.

Now, we apply the test to all of them.

Of the 100 who are positive, the test will correctly return positive for 99 of them, and get one wrong returning a negative.

Of the 99,900 who are negative, the test will correctly return negative for 98,901 of them, and will get 999 wrong returning a positive for them.

You got a positive result. That means you could either be in the first population of 99 where the test correctly returned positive. Or you could be in the second population of 999 where the test wrongly returned a positive.

What is the probability that you are in the first population that is actually positive? You can calculate the answer easily now - 99 / (99 + 999) = 9 % chance you are actually positive.

So you take a test which is 99 % accurate. The test result is positive. The chance you are actually positive is only 9 % (assuming spread in the overall population is 0.1 %).