## June 12, 2009

### The Monty Hall Trap

“Suppose you are on a game show, and you are given the choice
of three doors: Behind one door is a Rolls Royce car and behind
the others, goats. Let's assume that you would prefer a Rolls Royce
to a goat. You pick a door, say No.1, and the host, who knows
what's behind the doors, opens another door, say No.3, which has
a goat. He then says to you, "Do you want to pick door No. 2?

XZ said...

Yes, i'll pick door No.2 and take my Rolls Royce thank you. =)

Kiklibio said...

Don't change doors.
In the beginning when you pick a door u have 33,33...% chance of winning the rolce.
when they open another door with goats behind it, you gain 33% on top of your previous 33%
so now you have 66% chance of winning the rolce.
If you change doors you will have less.

correct? (thought it was likt this, but might be mistaken

XZ said...

No... i think it's if u change the door u'll get 66% chance istead of sticking to your 1st choice which would still be 33%.

Damien said...

It is always to your advantage to switch doors. At the start you have a 33.33% chance of picking the right door. The host always opens a door that shows a goat, so if you switch you greatly increase your odds of winning since you would only lose if you chose the car door to begin with.
Ex: (the car is behind door #1)
case#1: you pick door #1, host shows door#3, you switch and lose
case#2: you pick door #2, host shows door#3, you switch to door #1 and win.
case#3: you pick door #3, host shows door#2, you switch to door #1 and win.

Secret Squïrrel said...

This is a very well-known problem and, despite being completely analysed, still confuses a lot of people. XZ is correct (although he won't get the car if it's behind Door 1) and Damien's explanation is good but I'm sure that some people will still have some doubt.

Another way to look at it is that since there is only a 1/3 chance of you picking the correct door, there is a 2/3 chance that the car is behind one of the other 2 doors (1/3 each door).

When the host shows you that (say) Door 3 does not have the car, the chance of it being behind the door you originally chose is unchanged (still 1/3) AND the chance that it is behind one of the other 2 doors is also unchanged (2/3). However, you now have new information that the chance of it being behind Door 3 is 0/3, so the chance of it being behind Door 2 must be 2/3.

You should always change - two out of three times it will be to your advantage. You are unlikely to get better odds than that!

Maria said...

You have an equal chance with both of the remaining 2 doors.

You have a 1/3 chance of picking the right door first time around.

After the host shows you that door 3 is incorrect there is a 1/2 chance it is behind either of the remaining 2 doors. So you have an equal chance with either door.

This is because the opening of door 3 is an event that has happened. Whatever the probabilities were before are irrelevant. Now there are 2 doors and only 1 rolls royce.

Secret Squïrrel said...

@Maria

As I said in my above post, this is a well-known problem but it still causes confusion for some people.

Rather than me re-iterating what has already been said, you might like to do a little reading yourself to see why you are mistaken.

http://en.wikipedia.org/wiki/Monty_Hall_problem
is a good place to start.

guilda2 said...

Like Maria's.
Beside, The host could be tricking us by openning the wrong dorr, since he knows what's behind each door. So I don't want to change.

pannag said...

To answer Maria's and Guilda's doubts:
Lets exaggerate the problem. Say there are a million doors, and I ask you to pick one. There is a very little chance that you would pick the right one. Very high chance that the car is behind one of the remaining 999,999 doors, right?
Now, if I open 999,998 other doors, and none of them has a car, then there is only door left out of 999,999. Would you change or not? The answer is yes. It is like choosing 999,999 doors against one.

Secret Squïrrel said...

@pannag, Good one!

@guilda2, The host will only ever open a door with a goat behind it, so it will always be the "wrong" door. If he opens one with a car behind it, you'll see it and will probably change, although I'm starting to think that maybe there's still someone who wouldn't change because they suspect a trick. I hope they like goats.

Anonymous said...

It doesn't matter what door you pick because no matter what you choose, the Host will just open the door without the car.