Well, I'll admit I'm mathamatically challenged. I've no problem with logic, just formulas, so I'm going to be difficult.

When you choose the first cup, you have a one in three chance of picking the right cup.

When you choose a second time, you have a one in two, or 50% chance of being right. Forget the third cup - you're making a choice between 2 cups, and it makes no difference if you pick the same cup again - it's still a choice. If it is the one with the prize, it will have survived a 1/2 chance because you made a second choice.

I don't think adding up the probablilities makes sense. I think that taking one of the cups away just changes the number to pick from. You can't have a 2/3 chance of winning if there are only 2 cups.

I wonder if anyone's done experiments...