"Deal or No Deal" math

[Hopey]

I was out for lunch today with some of my co-workers and they were having the following debate:

-You are playing "Deal or no Deal"
-All of the briefcases except for one (and the one you chose initially) have been chosen.
-The two dollar amounts left are $1 million and .01.
-You are given the choice to keep the briefcase that you chose initially, or to switch it with the briefcase that is left on stage.
-What do you do?

I argued that it doesn't matter what you do, you have a 50/50 shot at the $1 million, no matter which briefcase you decide to open. However, a couple of guys were arguing adamantly that you should always choose the briefcase that is left on stage. I argued that the fact that you moved one of the briefcases off the stage initially does not affect the contents of the briefcase, and does not affect the probability that it contains $1 million.

Can someone please put this issue to rest? I feel like an idiot for asking this question, but no matter how much I argued, they kept insisting that I was wrong.