The Monty Hall problem is a probability puzzle based on an old US game show. You are shown three doors. One contains a car, while the other two contains goats. The game show host invites you to pick a door. He then opens one of the doors containing a goat and asks, "Do you want to stick with your original selection or switch to the remaining door?"
What should you do…assuming that you don’t wish to own a goat?
I’ll take all the fun out of it. You should switch. It doubles your chances of getting the car.
Convinced? Probably not. Even when presented with the solution, many people struggle to accept it. It’s not particularly intuitive.
Martin Johnsson recently discussed this on his blog. He used paper simulation, computer modelling and mathematics to try and satisfy himself of the wisdom of switching. He concluded
…I’m not sure I have convinced myself of the solution to the generalised problem yet.
Reasoning about probabilities is hard. It’s very easy to be led astray by our "gut". As the eminent statistician Sir David Spiegelhalter has noted
…when asked a basic school question using probability, I have to […] try it a few different ways, and finally announce what I hope is the correct answer.
This dereliction of our instincts means that it’s essential to draw on the formal methods of statistics and Monte Carlo simulation when making important decisions.