# Riddle me this

Doc Al
Mentor
Njorl said:
No it won't. You are erroneously assuming that the revelation that there is no prize behind the opened door does not affect your chance of winning.
Njorl, I think you are misinterpreting Integral's comment. I believe Integral would agree that revealing the door without a prize provides essential information.

Njorl
Doc Al said:
Njorl, I think you are misinterpreting Integral's comment. I believe Integral would agree that revealing the door without a prize provides essential information.

Sorry, I deleted my message, wasn't fast enough.

"I think I better think it out again!" (Fagan from Oliver)

Njorl
I was thinking about the problem incorrectly.

If the host opens a door that is guaranteed to be empty, you are better off switching. If the host opens a door that may or may not be empty, it does not matter.

It becomes easier to visualize with many doors.

We will play the game two ways.

The first way, the host opens random doors that are not your choice. In this game, it is anti-climactic because the prize is almost always revealed well before it comes down to two. It is never beneficial (or harmful) to switch choices in this game. Even if we are reduced to two doors, it does not matter, but there is only a 1 in 500 chance to get to this point.

Essentially, you and the host are making 1 in a 1000 chance guesses. Trading your guess for his is pointless.

In the other game, which simulates our logic problem, you make a 1 in a thousand guess. The host, by eliminating all possibilities but two is then asking you "Do you think your guess is right or do you think your guess is wrong?" Since your guess was 1/1000, it was probably wrong.

Integral
Staff Emeritus
Gold Member
I didn't see Norjls post, but it is my understanding of this riddle, that the host has knowledge of where the prize is and always exposes an empty door.

Otherwise the host will expose the prize 1 out of 3 times.

Njorl
Here is a good way to think of it. Choose a door. The host then asks you if you think you guessed right or wrong. Since you only had a 1/3 chance to be right, you should say that you were wrong. Saying you answered wrong is equivalent to switching doors after one was revealed. While the final choice between two doors might appear to be a standard 50/50 chance, it is not. It is a choice between whether your original guess was right or wrong.

If there was a chance that the host would reveal the prize when he opened that first door, then you would be justified in saying it doesn't matter

I must admit, my convictions were strong in the opposite direction at the onset. So many people are bad at probability that my first instinct was to show those ignoramuses how they just don't know math. Oops

Njorl

the door on the RIGHT!!!

When I first encountered this problem, I did't seem right to me that switching would help. Many posters before me gave other excellent ways of looking at the problem that might convince one that switching is good. I will propose another way of looking at the problem here that doesn't rely much on the probability of it all, but on intuition.

Consider these two games:

Game 1: You get to pick one of the curtains, the host doesn't reveal anything to you, and you are given the option to switch.

Game 2: This is the original game. You pick one curtain, the host reveals an empty curtain, and you are given the option to switch.

What happens if you don't switch in either of the two games? Your intuition should tell you here that your chances of winning will be the same in both games.

Now what happens if you do switch? Which game will you win more in if you play the game a considerable amount of times? (Note that in Game 1, your chances of winning will not change whether you switch or not).

Before I leave, I would like to propose a variation of the original game. Suppose a contestant picks one of the three curtains and leaves. The host lifts one of the 'bad' curtains leaving two curtains (one of them being the one the contestant picked). In comes another contestant (who doesn't know what has happened) and is asked to picked a curtain. The probability that this contestant will choose the winning curtain is 1/2 right? Now what if the contestant chose the the same curtain as previous one. Would his/her chances of winning increase by switching? What if before choosing one of the two curtains, the current contestant was told the whole story of what happened. Will this information allow the current contestant to choose a curtain that will increase his chances of winning?

e(ho0n3

jcsd