# The Monty Hall problem

It’s

and here’s your host

!!! MONTY HALL !!!

In Monty Hall’s game show you are the contestant.

In this game there are three doors to rooms at the back of the stage and you are told that
behind one of them is a prize and the other doors lead to empty rooms.

You are asked to choose a door. If you pick the door with the prize, you win it,
if empty you lose. Then Monty, who knows where the prize is, opens an empty
door. He then offers you the chance to switch doors or stick with the one you first picked.
Should you switch? Answer is either YES, NO or IT DOESN’T MATTER. Hint: Remember it’s
a probability puzzle.

Answer.

The intuitive answer is that it doesn’t matter because the odds are equally 1 in 2 or 50 / 50
for each of the closed doors. However the correct answer is YES, you should change your choice
because you are twice as likely to win the prize. But why? Even mathematicians struggled with
this one.

At the start you have a 1 in 3 chance of picking the right door. But, when you see the
first empty room, it does not alter the odds to 1 in 2 for each door as seems logical
but 2 chances in 3 for the door not chosen.

There are two situations. You first choice was right or your first choice was wrong. In the
first situation you would be wrong to change your mind, in the second you would be right to change
your mind. However you are twice as likely to be in the second situation as in the first situation.
Here is a simple simulation to try out a few
hundred times either changing or sticking with your first choice.

The backstory.

Monty Hall pointed out the psychological factors which were a major part of his TV game show.
Opening a door puts pressure on the contestant who thinks that the odds on the chosen door
hiding the prize have gone up from 1 in 3 to 1 in 2 so it is sensible not to change. Also
Mr. Hall did __not__ have to offer a switch according to the rules of the show and he
usually didn’t offer it. He had another possible tactic.
Whenever the contestant began with the __wrong__ door, Mr. Hall promptly opened it and
the contestant lost; whenever the contestant started out with the __right__ door, Mr. Hall
allowed him to switch doors and lose! More bizarre he sometimes offered increasing amounts of
money to switch or not in order to jazz up the game.

The more elegant original puzzle was formulated by Joseph Bertrand in 1889. In this puzzle there
are three boxes, one holds two gold coins, one a gold and a silver and one has two silver coins.
After choosing a box at random and taking out one coin without looking, which happens to be a
gold coin, what are the odds that the other coin is also a gold one? The answer again is 2 in 3,
not 1 in 2.