Coloured Hats Riddle

[iago]

Yes, Coloured.

Anyways, this is an interesting riddle that caused a lot of argument at another forum I visit, so I thought I'd see who here can figure out the solution. Please don't Google it! Here's the problem:

You and two other people are brought into a dark room, and offered a chance to win $1,000,000. The problem is explained to you, and you are given time to plan with the others. A hat will be put on each of your heads. The hat will either be yellow or red. The light will be turned on, and you will see the others' hats. You will not be allowed to communicate in any way, and you are all allowed to guess the colour of the hat you're wearing, at the same time. If any person guesses incorrectly, you all lose. If one or more people guess correctly and the others decline to answer, you win the $1,000,000. If nobody answers, you lose.

To summarize:
- You are all given a randomly coloured hat, and an opportunity to see the others' hats
- Without communicating, you may guess at your colour at the same time as others
- If anybody is wrong or nobody guesses, you lose
- If one or more people are right, you win

Go!

[Sidoh]

Can the hats be all red or all yellow?

By the way, why is this in UNIX/Linux?

[iago]

Yes, the hats can be all red or all yellow.

It's in Unix becase "Unix" looks like "Humor" from work and with too little sleep. Moved.

[iago]

By the way, there is no way to get it 100%, which is logical. There's no communication or any way to see your own hat.

So the question is: what's the best strategy to maximize your chances of winning?

[Sidoh]

Yeah, I was thinking that.

Well, I suppose the most straightforward solution, assuming that the hat colors are chosen in a truly random way, is to have the person that sees two hats of the same color to say the opposite color (since there's a higher chance that his hat is that color).  If no one says anything immediately, assign one person to pick a color at random (I'm not sure that's legal).

[iago]

Yeah, I was thinking that.

Well, I suppose the most straightforward solution, assuming that the hat colors are chosen in a truly random way, is to have the person that sees two hats of the same color to say the opposite color (since there's a higher chance that his hat is that color).  If no one says anything immediately, assign one person to pick a color at random (I'm not sure that's legal).

That second part wouldn't be legal, but the first part hit the solution right on the head.

So just finish it off (yes, I'm making up rules): what is the probability of success using that strategy, and can you prove it?

[Newby]

It's in Unix becase "Unix" looks like "Humor" from work and with too little sleep. Moved.

I figured it was Unix because there are Red Hats. Heh.

[Sidoh]

25% chance that the hats are RRR or YYY: 2*(1/2)^3, which would cause the strategy to fail.

[Newby]

Well, I suppose the most straightforward solution, assuming that the hat colors are chosen in a truly random way, is to have the person that sees two hats of the same color to say the opposite color (since there's a higher chance that his hat is that color).

Or, even better! (Not sure if this is legal.)

If he sees two hats of the same color, have him "decline to answer." That way, they can look at each other, and know the color of their hat.

I guess that's communicating, but whatever. It's a plan.

Wait, they're at the same time. I am r-tarded.

[iago]

Haha yeah, Sidoh has the best possible answer. It'll succeed 75% of the time, whereas one person randomly guessing will get it right 50%.

The big issue on the other board was that the colours of the other hats don't dictate yours. ie, if they're both yellow, the chances of yours being yellow is still 50%. Which is an interesting (and incorrect) way of looking at it.

[Sidoh]

Yeah, agreed.  That's only true if you don't know anything about the colors of the other hats.

[Newby]

The big issue on the other board was that the colours of the other hats don't dictate yours. ie, if they're both yellow, the chances of yours being yellow is still 50%. Which is an interesting (and incorrect) way of looking at it.

'tis what I thought.

[iago]

The other good part about that strategy is if you're wrong, you're very wrong. If all three have the same colour, everybody guesses wrong at the same time. If you're going to screw up, screw up right!

[Super_X]

My idea is to look at the two hats, then trade with some one. Rotate it to the left. Then every one knows what one they had, and now have.
There's no rule agains that, from what I can see.