Statistical Questions: This Is One of My Favorites

I am not sure where I first saw this question.

For some reason though, it stuck in my mind and I never forgot it.

It is simple, and just takes a little bit of reasoning to figure out.

At the same time – nearly nobody can calculate the right answer.

Here it is.

Question

99% of the population is Flu free. 

You are tested for the Flu and the test is positive. 

The test is wrong 5% of the time. 

What is the percent chance you are actually still Flu free?

Answer

Out of a 100,000 population, 99,000 are Flu free. 

1,000 are infected.

Of the 99,000 Flu free, 94,050 had correct test results that said they were Flu free.  
4,950 of the Flu free received incorrect test results that said they tested positive.

Of the 1,000 infected, 950 had correct test results that said they tested positive.
50 of the infected received incorrect test results that said they were Flu free.

In all, 5,900 people got the bad news that they have the flu.  

However, of this 5,900, only 950 actually carry the flu.

4,950 / 5,900 = .84 = 84%

After testing positive for the Flu, there is still an 84% chance you are Flu free.

This is because 99% of the population is Flu free, and chances are – you are part of the majority.