Monty Hall Three Door Problem

The three door Monty Hall Problem is so called because it is based on a game show in which the host (Monty Hall) shows three doors to the contestant and asks them to pick one. Behind one of the doors is a car and behind the other two is nothing. Once the contestant has picked a door the host opens one of the other doors to show that there is nothing behind it and then asks if the contestant wants to swap.

This app replicates this puzzle and lets you choose a door and then allows you to stick or swap. It calculates the probabilities based on the choices you make to show that you have a much greater chance of winning if you swap.

The maths behind this are a little counter-intuitive as at first glance it doesn't seem as though swapping should give you an advantage but it does. Initially you have a one in three chance of guessing the winning door but this means that there is a two in three chance that the car lies behind one of the other doors (and a 100% chance that one of the other two doors has nothing behind it). Just because the host shows you that one of the other doors contains nothing, as it has to, doesn't change the probabilities (you still will only choose the correct door first time around one in three times and a two in three chance that the car is behind one of the other two). Effectively if you swap you are getting what is behind the other two doors, one of which you know from the start has to be nothing and the other may be a car or may be nothing.

Swapping could lose you the prize but if you repeat the puzzle many times you should see that the probability of winning approaches 33.33% if you don't swap and 66.67% if you do.

There's no prizes in this app, it just calculates the probabilities of a win.