What's Behind the Doors? Exploring the Probability Dilemma in the Game of Chance

[tanujkush]

Welcome to the game of doors!

Anchor: Alright then, you have three doors in front of you, A, B, C. Two of these three contain nothing, one contains the proof of the unified theory yet unknown to man . You can choose anyone door, which one do you choose?

You: Uhmmm.. I think I'll go with door A

Anchor: Are you sure?

You: Yeah!

Anchor: (opening door C) Well well, there is nothing behind door C! Arent you glad you didnt choose that door?

You: (wiping the sweat off) Yeah! Now show me what's behind door A.

Anchor: Wait a minute, what if I made you an offer. Would you like to change your door? Would you like to choose door B instead?

You: (Damn, I should have paid attention to those undergrad probability lessons!) Uhmmm.. I don't know... I think I will go with what probability tells me. Let's ask the folks over at Physics Forums what they think? Should I change doors or stick?

 

[CompuChip]

Change.

(CompuChip wonders why this puzzle has been around since like 1000 AD and nobody got to see the proof yet).

 

[tanujkush]

Change.

(CompuChip wonders why this puzzle has been around since like 1000 AD and nobody got to see the proof yet).

there is a proper closed form proof for this really.

 

[lanedance]

Ch-ch-changes
Just going to have to be a different man
Time may change me
But I can't trace time

 

[CompuChip]

there is a proper closed form proof for this really.

Then you should publish it and get famous.

(I was talking about the Riemann hypothesis of course, the proof of why changing the doors increases your odds of winning to 2/3 can be written out on a single line).

 

[tanujkush]

(I was talking about the Riemann hypothesis of course, the proof of why changing the doors increases your odds of winning to 2/3 can be written out on a single line).

I of course, was talking about the second part of your quote.

 

[Rogerio]

...
Anchor: Wait a minute, what if I made you an offer. Would you like to change your door? Would you like to choose door B instead?

It depends.
Does the Anchor know the doors a priori?
If he doesn't know what is behind each door, then changing from door A to door B makes no difference.

 

[HallsofIvy]

I remember, about 20 years ago, seeing this problem as an exercise in chapter 1 of an introductory probability text.

 

[EnumaElish]

This problem still stumps me somewhat, philosophically speaking. Suppose the anchor did not know a priori, but I didn't know that she didn't know. Shouldn't I still change?

Put differently: suppose I don't know with certainty whether she knew. If she knew, then I better change. If she didn't, then (by the logic presented somewhere along the thread) it's all random and I'll be no worse off if I changed. On the net, I should change.

Since I can never know with certainty that she didn't know, I should change every single time I am in this position.