Fri Jan 13 21:29:41 EST 2006 (Updated 4/21/24 after input from Rodrigue Rizk)

Here's the so-called "Monty Hall Problem".  You are playing a game
with Monty Hall.  In the game, there are three doors.  Behind one is
money and behind the other two, goats.  (For the benefit of people
like my children, the goats should be considered to be less valuable
than the money.)

You pick a door. Monty now must now use his knowledge of the location
of the money and open one of the other two doors to reveal a goat. You
now have two strategies:

 1. Stay: Stick with the door you originally picked.
 2. Switch: Change your choice to the door that Monty did not open.

Which strategy is better?  Many people have a strong intuition that
these two strategies are equivalent.  However, the switch strategy is
actually twice as likely to be successful.

I first heard about the problem many years ago.  I was excited that my
spouse was introduced to it in a class she is taking right now.  She
told me that she didn't understand the explanation, so I came up with
one that I think is a good way of understanding how it works.

After you've picked the door, let's say you have two *other* strategies:

 1. Ignore: Don't ask Monty for help and keep whichever door you first picked.
 2. Mercy: You ask Monty for mercy.  Monty uses his knowledge of the
    location of the money to do the following.  If you originally
    picked a door with a goat, Monty takes pity and shows you where
    the money is. On the other hand, if you asked for mercy when you
    had already picked the money, Monty gets annoyed and takes it
    away, giving you a goat instead.  

Which of these two strategies do you prefer?  I think it's clear that
the mercy strategy is the winner---you get the money 2/3 of the time
(those times when you initially chose wrong). The ignore strategy has
a 1/3 payoff, assuming things are truly random.

So, the "aha" insight is supposed to come when you realize that
"ignore" is really "stay" and "mercy" is really "switch"!

-Michael Littman