There's been a few going around lately... here's my contribution.   This one's a classic.

You're lucky enough to play in a game show, and it's down to the final question.  There are 3 identical looking boxes in front of you, and one of them contains the grand prize.  The host now tells you if you pick the right box, you win the grand prize.  If you pick one of the other two... you go home empty handed.

Seeing there's nothing to choose between the 3, you go ahead and just randomly pick one.  Now, the host says, "I know which box contains the grand prize, and I'm gonna help you out a bit."  Out of the 2 remaining boxes, he opens one of them, and reveals that it's empty.

Now he asks you, "Do you want to change your pick to the other unopened box?  Or do you just want to stay with your original pick?"

What do you do?  (assuming you want to WIN the grand prize, of course) 

(Also, the host is not intentionally trying to trick you.  That's the way the game is played.)

you pick the one on the far right

sorry, this one is too classic. you have to change, because you had a chance of 1/3 to pick the right box at the beginning. after he reveals one wrong box, the remaining box contains not only it's own third chance to be the right one, but also the third chance of the other box, which makes it 2/3.

Probably this is too complicated, so I'll show all possible situations:
(x=you, A/B/C boxes, assuming the price is behind A)

1. Situation:

A X
B
C

The moderator either opens B or C. Here you loose if you change, because you were standing on the right box.

2. Situation:

A
B X
C

The moderator has to open C because you are standing on B and A contains the price. You win if you change.

3. Situation:

A
B
C X

Same as above, just with different letters. He has to open B and you win if you change.

So you win in two of three situations, which means changing gives you a chance of 2/3 to win while staying is only 1/3.

I remember this from watching Numb3rs! Cars and goats!

Yeah you switch cause the announcer has a limited choice, he can only pick one that doesnt have the item. Since his choice is constrained you get better odds. An easy way to look at it is if there were 100 boxes and you picked 1, then the announcer says I will open 98 that don't have the item, then do you want to switch. The choice would be very obvious.

You pick the right box. lol

The choice isn't obvious... or else when it first came out in the 60s or 70s it wouldn't have been controversial.  Even some math professors specializing in probability got it wrong.   The columnist who wrote it received hundreds of letters from readers around the country saying that she has "No understanding of probability" , "Needs a basic course in Math" , "Misleading the American public".  Among them was a UPenn Math professor!  Later, he wrote her asking her not to reveal his identity because he is utterly embarassed by it.

It's amazing how hindsight is always 20/20. 

To those who haven't seen it before, the question you should be asking yourself is: why aren't the remaining choices both 50/50?  After all, you're picking one of two, so it's a 50% shot.

What, I'm confused... completly, why wouldn't be a 50/50 chance? If you pick the right one first, he's got the choice of two empty boxes to open, so he opens one. You switch, and you loose You stay, and you win But you're all saying that you have to change? What is this crazy voodoo?

If you said the rules right, you would have already lost with your first choice if it was wrong. As you didn't loose, it must be right so why would you change?

read my post, Samuel. The box where you are not standing unites the third chance of itself and the third chance of the revealed box.

and jlauro, you haven't read the rules right I think...