There are always different ways of looking at these problems. Suppose you don't actually pick a particular cup, but visualise that there is a 1/3 chance of any cup being the one. When the house picks a cup there is a 1/3 chance they will pick the right one. There is therefore a 2/3 chance that the remaining two cups hold the prize - i.e., 1/3 each.

Whether you actually pull one out initially or not is irrelevant - after all - all you are doing is putting your hand on it. The house still has exactly the same chance (1/3) no matter what cup they pick. Once the house eliminates a 1/3 possibility by finding an empty cup, there is 2/3 possibility remaining - and this is equally divided between the two cups.

The chance of someone else having MY birthday is 1/365. The first person's birthday is always given. This is not the same as having two separate bags of 365 people, each with different birthdays, and separately drawing a person from each bag at random. The probability of both people drawn having a PARTICULAR birthday is 1/365 x 1/365 (in each case there is a 1/365 probability of the person having a particular birthday). However, if you now say that, whatever birthday is drawn from the first bag it is right, then the probability of matching improves to 1/365. This is because the probability of the first person having ANY birth date is 1.

Chris