Do you see blue?

 
Imagine that a town has only 2 colours of car: 85% are blue and 15% are green.  A person witnesses a hit-and-run and says that he saw a green car.  If it is known that witnesses identify the colour of cars correctly 80% of the time, what are the chances that the car is actually green?
 
Did you look at this question, have a smirk (or, ok…a smile) and then say, “Caught you!  The answer is 80%” ?
Well, I did.
 
This is a fabulous example of a fallacy in statistics (and in making decisions) called The Base Rate Fallacy: my brain ignored all the statistical information (or base rate) and focused instead on specific information on the witness. 
 
So, let’s solve it:
If we randomly selected 100 cars, 85 would be blue and 15 would be green, right?
Since the witness gets the colour right 80% of the time:
-       he will identify 12 cars correctly (80% of 15 green cars)
-       he will also identify 17 cars wrongly (20% of the 85 blue cars!)
 
So, his error rate is a massive 17/29, which is 59%
…and the hit rate is just 41%.
 

The base rate fallacy can show up in our understanding of many issues (or, to be more precise, our lack of understanding of many issues) including infections.