The extrapolation to 1000 cups helped me too. The event is divided in two stages. In the first, you have a probability of 1/1000 to choose the prized cup. The house is OBLIGATED to take 998 cups out and leave to you the alternative: "Did you win in the first stage, with 99.9 % probabilities against you, AND will win AGAIN with 50% against you, in two consecutive choices ? Or did you loose first, with 99.9% probability of loosing, and will now win with this 50%, in two consecutive choices ?"

Undoubtedly, and very intuitively, I would change !

Moreover, I imagined a situation in which the house is allowed to change places of the prize before you make your second choice. This would become TWO games, one with 33.3% probabilities of winning, and another one with 50%. Since the house cannot change the place of the prize, and always disclose all remaining cups minus one, it is giving you a very valuable information that changes your probability to win in the second stage.

You can be sure that if I open a casino, I'll hide out the three cups for a moment before you attempt your second choice.

Un abrazo - Andrés (and thanks to all for taking the trouble to explain this so well )