Unlocking the Mystery: The Science Behind the Game Show Brain Teaser

[island-boy]

Okay, you're in a game show that is akin to Deal or No Deal.
You are given 3 suitcases to choose from, 1 of which contains a million dollars, the other two are empty.
Once you have chosen a suitcase, the host (who knows beforehand which of the 3 suitcases contain the million dollars), opens one of the (remianing unchosen) suitcases of which he is sure is empty. You are then given the choice of switching the suitcase you chose with the other suitcase that is unopened.
Prove that it is beneficial to switch suitcases.

I am quite confused by this...I'm thinking that switching cases would give you a 50% chance of getting the million dollars, while not switching will only give you a third of a chance. But this does not make sense intuitively.

help please.

 

[NateTG]

This is the infamous 'Monty Hall' problem. (That's the name of the gameshow host.) You may be able to get more information by using that on google.

Alternatively, what would it be like if there were 1000 suitcases, you pick one, and then monty opens 998 empties. Does it make sense to switch then?

 

[matt grime]

The key is that the game show host knows which are empty. By opening them he hasn't changed the probability that you picked correctly in the first place, which was 1/3, so switching will always make sense.

Nate's 'thought experiment' is in my opinion the best way to think about it, and here is another.

Suppose instead that the boxes contain 1 pound, 1000 pounds and 1,000,000 pounds, you pick one and the host opens which ever of the two boxes has the least amount of money in. (This serves to differentiate all three boxes for you.) Now, what are the options? If you pick the one pound box, then switching gets you 1,000,000, if you pick the 1,000 pound box you also get a million if you switch, and if you pick the 1,000,000 pound box you get 1,000. if you switch. So switching is in your favour 2 times out of 3 hence you should do it.

 

[island-boy]

hey guys, thanks for the help, I was able to get the answer,
I also found some useful discussions on this brain teaser through google:
http://www.deanesmay.com/posts/1105028800.shtml
http://patterico.com/2005/01/07/2547/fun-brain-teaser/

 

[mbrmbrg]

Wow.

The patterico site brings up the issue of whether or not it matters if "Montry" knows which door is empty, or if he just picks an empty one (I think of it as the empty doors are unlocked, the prize door is locked. You close your eyes, and Monty tries the doors, opening the first one he can that you didn't pick).
I say it doesn't matter if he knew beforehand, but then again, I was also convinced that switching to begin with was pointless.