Cup and ball game: what are the odds of that?

#1
OK we are having an intelectual discussion (no, really) about the change in odds during the classic cup and ball game. To explain:

  • a dealer has three cups under one of which is a ball
  • you pick one
  • the probability at this stage of winning is 1 in 3
  • the dealer then guarantees they will remove an empty cup

If the "experiment" remains unchanged the probability at this stage, as I understand it, is 2 in 3 as your cup is one of the two remaining cups which must have the ball underneath.

If, however, you change your mind (equivalent to a post-hoc analysis) then you change the game and essentially the probability becomes 1 in 2.

Can someone with a bigger brain than mine confirm or deny this?

Thanks.