Monty Hall’s Perplexing Present

Before you read on to find out why this is the case, have a play with the game at https://math.ucsd.edu/~crypto/Monty/monty.html 

Or find some objects around your house to be your “present” and “doors” and play for yourself. Try sticking with the same door sometimes and switching other times. Do you win more often if you stick or if you switch?

We’ll continue thinking about what happens if you pick door 3.

There are three places the present could be – door 1, door 2 or door 3!

If the present was behind door 1, I would open door 2.
If the present was behind door 2, I would open door 1.
If the present was behind door 3, I would open either door 1 or 2.

Each of these has a 1 in 3 chance of happening – since the present could be behind any door.
There are two situations where I would win by switching and only one where I wouldn’t!
So, since these all have a 1 in 3 chance, I have a 2 in 3 chance of winning if I switch and only a 1 in 3 chance of winning if I don’t.
So, it’s clear I should switch!