# Monty Hall

by Kevin J. Lin

You should switch.

The "Monty Hall" problem, as this puzzle is usually called, is probably one of the most heavily debated puzzles of all time. Not only does it appear almost weekly in "rec.puzzles", it has been tackled by Marilyn vos Savant in her column "Ask Marilyn" and by Cecil Adams in "The Straight Dope. Almost everyone who hears the problem and the correct solution doesn't believe it. It seems so counter-intuitive.

### How Could Switching Possibly Improve Your Odds?

Here's the simple explanation (we'll get down to details later):

What is the probability you picked the correct door on your first guess? This is easy- one in three.

What is the probability the host will open a door with a "bad prize" *other than the door you picked* (in our story the chocolate bunny)? This is 100%, as defined in the problem.
So there's a one in three chance the gold bunny is behind the door you chose, and a zero in three chance it is behind the door the host picked. Therefore, there must be a two in three chance it's behind the only other door.

Seems strange, doesn't it?

The host, in effect, is giving you a hint where the prize is. He's providing information about the state of the system which you did not have before. It is important that the host's intentions never be a relevant factor- as we described the problem, the host does not have the opportunity to try to trick you even if he wanted to. Every single time he must open a door you did not choose, and reveal a chocolate bunny.

While on the subject of the host's predictability, let me bring up Marilyn vos Savant's answer to this puzzle. She posted an answer in her column back in 1990, and thousands of disbelieving readers sent in letters claiming she was wrong.

Marilyn did NOT err in the answer. She erred in the question. She made her response based on a wording of the question sent in by a reader, which made no indication that the host would behave the same no matter what door you selected. Without this critical bit of information, the puzzle is unsolvable. Here's the letter she was replying to:

Suppose you're on a game show, and you're
given the choice of three doors. Behind
one door is a car, behind the others, goats.
You pick a door, say number 1, and the host,
who knows what's behind the doors, opens
another door, say number 3, which has a
goat. He says to you, "Do you want to
pick door number 2?" Is it to your
advantage to switch your choice of doors?
Craig. F. Whitaker
Columbia, MD

Can you see why this wording is ambiguous? It only describes one particular iteration of the game, with no guarantee that the host must always offer you the chance to switch.