Amos, you're right, and I'll try to explain below.

Catspaw, sorry, but I don't understand you. Who or what is Monty?

Jon and Jeri, you've given the most intuitive answer - that at the time of your second choice there are only two cups in the running, and you don't know which, so you think it's a 50-50 thing. But this is not so.

In fact your best bet is to switch cups. If you switch, you have a 2/3 chance of winning, but if you don't switch, it's only 1/3.

I know this is really counterintuitive, so I'll try two different ways of explaining it.

One: remember that if you guessed right the first time, switching will definitely make you lose. If you guessed wrong the first time, switching will definitely make you win, since the other wrong cup has been eliminated for you. The probability of guessing right the first time is 1/3, so the probability that switching will make you lose is also 1/3. The probability of guessing wrong the first time is 2/3, so the probability that switching will make you win is also 2/3.

Two: think of it this way: when you make your first guess and point out a cup, what you are really doing is dividing the cups into two groups: a small group with one cup, and a big group with two cups. When you are given a second chance to guess, what you are really doing is saying whether you think it's more likely that the prize will be in the big group or the small group. It's more reasonable to bet that the prize will be somewhere in the big group. When the house lifts an empty cup, that just shows that at least one of the big group is empty, but you knew that already, so that's not really relevant to the question of whether the big group or the small group is a better bet (although it is useful information because it tells which member of the big group would have the prize if one of them does). There's a 2/3 chance that the prize is in the big group, so your chances are better if you switch over to the big group.

I know this sounds terribly convoluted compared to the beautiful simplicity of "two cups, the one you picked and one not touched yet, so 50-50 chances", so if you're still not convinced, try this mental experiment:

Imagine you sat down with a ridiculously patient friend to play this game 1000 times. The plan is for your friend to switch every time, then you'll see if the number of times he wins is closer to 500 or 667. Work through in your mind what would happen. He would pick an empty cup the first time approximately 667 times, assuming that his guessing and your prize-hiding were random. So he would win approximately 667 times.