NB: for the following, I am referring to the "You pick a door, the host then always opens one of the two other doors, which is always a losing door, and you're always given the opportunity to switch" problem, which is the classic Monty Hall problem.
The first step when you're trying to convince someone that an intuitive answer is wrong should be to disprove the intuition. Showing the proof of the correct answer should be second.
In this case: the intuition is "there are two doors, therefore 50/50". The maths says "If there are n indistinguishable options, the chances are 1/n"... but the doors aren't indistinguishable... one of them the host couldn't open because you've chosen it... the other the host could open, but elected not to... possibly at random, but possibly because it's the winning door. So the doors are distinct, and a such, they're not necessarily 50/50.
To make the 1/3 result more clear, imagine that the doors are able to be physically moved around (I often use an analogy with Deal or No Deal here, and the briefcases). The door that you choose, you take with you to your side of the room, the two you don't choose I keep with me. The chance that you have the prize is 1/3, the chance that I have the prize is 2/3... this much is clear, since the doors are still indistinguishable (the only distinguishing feature is that you picked the one that you did, and you made the choice without any information).
Now, I open one of my doors and show it's the loser. This doesn't give any new information with respect to which side of the room the prize is on, since you know that one of my doors is, indeed, a loser.
So the chance you have the prize is still 1/3, and the chance I have it is 2/3. In the 2/3 chances that I have the prize, however, it is always the door I haven't opened, so there's a 2/3 chance that the prize is behind my closed door - you'll have a 2/3 chance if winning if you switch.
It is as EvanED says... if you choose wrong in the first place, which you have a 2/3 chance of doing, then switch, you will always win.
However, Yakk's gripe is very valid, hence the note at the top of my post.
While no one overhear you quickly tell me not cow cow.
but how about watch phone?