The Taxi-cab Problem
A cab was involved in a hit and run accident at night. Two cab companies, the Green and the Blue, operate in the city. 90% of the cabs in the city are Green and 10% are Blue.
A witness identified the cab as Blue. The court tested the reliability of the witness under the same circumstances that existed on the night of the accident and concluded that the witness correctly identified each one of the two colors 80% of the time and failed 20% of the time.
What is the probability that the cab involved in the accident was Blue rather than Green knowing that this witness identified it as Blue?
The Taxi-cab Problem
80% - Congratulations, like most people, including me, you’ve been fooled by your intuitions and got this completely wrong. Even though it doesn’t feel like it, what the witness saw is only a part of the picture.
Us humans are very bad at looking at the big picture of things. That’s why most people, including me, think 80% is the right answer when they first read about this problem, while the actual answer is 30% (we’ll see why soon).
But why are we even talking about this?
This problem shows how people tend to ignore very important information when they look at the world, and when we get the world wrong, we make bad decisions.
Let’s pretend (for a moment) there was no witness. (sounds sketchy I know)
The original problem had a city with 90% green cabs and 10% blue cabs.
To make it simpler let's look at a small town with just 9 green cabs and 1 blue cab.
But 9 and 1 is boring, go ahead and move the line to, I don’t know, 6 blue and 4 green cabs.
Welcome to our small town,
let’s call it Taxidentville.
The circles show the 10 places where an accident might happen.
Gladly, only one accident will happen.
Good, but which cab had the accident?
Like the accident, let’s place the cabs randomly.
(you can still change the number of green and blue cabs)
Blue cab accidents:
Ah ha! A cab had the accident!
But remember, this is only one possibility of many.
As we can see, when there’s no witness, counting the blue cabs is the best guess we have.
Our witness is only correct some of the time.
We simulate better or worse witnesses with the new slider at the top right. (it’s currently on 60%)
The eyes above the cars show us what the witness would see.