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Monty Hall Problem - not necessarily 2/3

It is a common mistake to automatically declare that the solution to the Monty Hall problem is 'switch for a chance of 2/3'.

The answer depends on the exact definition of the action that Monty takes. Specifically, whether Monty intentionally opened a goat-door. In the version given on the site the question does not say which is the case (although the given answer assumes he does).

If Monty knows where the prize is and intentionally opens the goat-door then the answer is indeed 2/3, as stated.

However, if Monty opens one of the other two doors in random and it just happened to be a goat, then the probability remains 1/2.

This is because finding a goat is more likely if the 2 doors have goats (100% chance), than if only 1 of them has a goat (50% chance).

Using Bayes Theorem we find that the preliminary probability (2/3) changes to 1/2 after incorporating this new evidence.

Saar Wilf
Wednesday, February 16, 2005

he has to open a door with a goat:
"monty then counters by showing you one of the doors with a goat behind it"

but if you change the rules so that he can open any door it's a totally different problem and you are right then the chances are 50/50.

tobi
Thursday, February 17, 2005

Another possibility is that, if you've chosen a goat door already, Monty opens your door and says "Congratulations, you win a goat!"

The point is you have to know in advance if Monty is strictly following a script (always opening a goat door and offering a choice), or if he's maliciously trying to maximize the player's chance of winning or losing.

I remember an interview with Monty Hall where he played the game with the interviewer.  He mixed up his strategy a lot, and the interviewer wound up winning far less than the 1/3 he should have won by random chance.

Anonymous
Monday, February 21, 2005

The ambiguity lies in whether Monty chooses to open a goat door or has to open a goat door.

If Monty has to open a goat door (which was not the case in the actual game show), then your chances when switching are 2/3.

If Monty can choose whether to open a goat door, then depending on his mood, the switching strategy's effectiveness ranges from 0 (in the worst case where he only offers you a choice to switch when you've selected a prize door) to 2/3 (in the best case where he always opens a goat door and allows you to switch). It all depends on whether Monty is feeling nasty or nice that day.

Free will is a tricky thing indeed.

WanFactory
Monday, March 07, 2005

for good info on why 1/2 is wrong and why 2/3 relies on the unstated assumption that monty MUST open a goat-door:

http://www25.brinkster.com/ranmath/marlright/montynyt.htm

WanFactory
Wednesday, March 09, 2005

I realize that Mr. Hall can manipulate the game whatever way he wants, according to the article. 

But, to get a probability of 1/2 you are assuming that Mr. Hall does not always open a goat door.  Do you mean that he sometimes opens a prize door?

Contestant: I'll take door number one.

Hall: Before we open door one, lets take a look at door number two.  It's a... car.  Ok, so I guess you did not win the car...  Folks, we'll be right back after these messages.

Jason Robbins
Wednesday, March 30, 2005

Not exactly, something like:

Contestant: I'll pick door number one.

Hall: Lets open door number one, gee that's too bad - you just won a light bulb. The car was behind door number two.

WanFactory
Monday, April 04, 2005

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