Department of Mathematics, Statistics
and Computer Science
Wim Ruitenburg's Fall 2004 MATH025.1001
The Three Doors Problem
This problem is also referred to as the Monty Hall problem.
There is a situation sketch is the book, pages 10-11.
The game host shows us three doors, behind one of which there is a big prize.
We are asked to pick one of the three doors.
Once our choice is final, and the prize is behind the door, we have won.
Obviously, our chance of winning is one-third, or 1/3.
What happens with our chances when the game show host adapts the rules a bit?
Suppose we have picked a door, say door one.
Then the game host opens another door, say door two, and reveals that there is
no prize behind door two.
Now we are offered the options of sticking with door one, or switch to door
three.
Should we switch, or should we stay?
What are our chances for door one, or for door three?
- Here is the standard answer.
There is a 1/3 chance that the prize is behind door one.
There is a 2/3 chance that the prize is behind door two or behind door three.
So although the show host has ruled out door two, the prize is still behind
doors two or three with chance 2/3.
The difference is that we also know that door two covers no prize.
So the chance for the prize being behind door three is 2/3.
So switch to door three!
- Many people have an intuitive resistance to the answer above.
This may be because of `hidden' assumptions.
For example, we assume that the three doors are not made of glass.
For otherwise we can see the prize through the door.
We also assume that the show host is telling the truth.
This and similar restrictions are usually perceived as obvious from the
context.
Other assumptions may be less obvious.
- Suppose the show host always makes the observation about there
being another door without a prize behind it, for all participants.
Then the argument above applies.
- Suppose the show host randomly makes the observation about
there being another door without a prize behind it, for all participants.
Then the argument above applies.
- Suppose the show host only makes the observation about there
being another door without a prize behind it, for participants whose original
choice was correct.
Then the above argument is obviously false: Never change.
- Suppose the show host makes the observation about there being another
door without a prize behind it, for the first time.
Then it depends on whether or not the show host is trying to trick us.
We may not know.
Let us change the problem in a seemingly irrelevant way as follows.
As before, we are asked to pick one of the three doors.
Once our choice is final, and the prize is behind the door, we have won.
Obviously, our chance of winning is 1/3.
Suppose we have picked door one.
Then a storm blows through the hall, and one of the doors is randomly blown
open.
Suppose that door two blew open.
Behind it there is no prize.
Now we are offered the options of sticking with door one, or switch to door
three.
Should we switch, or should we stay?
- Answer: It makes no difference.
The chance that the prize is behind door one is 1/2.
The chance that the prize is behind door three is 1/2.
A door flew open randomly, so the search has been reduced to door one and
door three, without preference.
It is only coincidence that the door with the prize did not fly open.
Last updated: September 2004
Comments & suggestions:
wimr@mscs.mu.edu