this is a well known problem. and the answer is:
**************** (answer below) ******************
the answer is switch, which gives u 2/3 odds. staying has 1/3.
why is it interesting? b/c a lot of ppl get it wrong. and *still* get it wrong after being told the answer. you can show them a chart with every possible initial setup and every choice you can make, and the results come out just as I said the answer is. but still some people think it's 50/50. including many mathematicians. according to my book on Erdos, this problem first got popular when the correct answer was given in a column, and 90% of the mail said she'd gotten it wrong, including from professional statisticians and such. a lot of the mail was angry. even Erdos (famous, brilliant mathematician) had trouble with it.
this is all very strange. the problem seems, to me, quite simple. lots of explanations have been given. here is mine that I think may get at the heart of why people are confused:
they think: there are 2 doors left it could be in, so it's 50/50. but lets look more closely at the part where the host reveals a door to you. what he's saying is IF your initial guess was wrong (which we know is 2/3 likely) THEN you should not pick this door here, b/c it has a goat. so he's telling you which one to pick (if your first guess was wrong). if your first guess was wrong, you now have a 100% chance when you switch doors, b/c you know which of the two has it. so you should switch banking on the fact your first guess is probably (2/3 likely) wrong, since you made it blindly from from 3 doors.