Your math is flawed. We have 50% to continue at each gate, but since we got kicked out if we failed, the chance we reach a floor is directly tied to the chance we reach the previous floor. Also, what Arinnaid calculated, is the chance we "can" reach a specific floor, from the beginning in one section. It will only stay 50% chance for every floor if we have 7 doors to choose at beginning and WON'T get kicked if we open the wrong door.
In regards to your gathering example, it is not the same. Think of gathering at 95% but if you failed a hit it close the gathering windows right away, even if you haven't finished all your attempts. That would be the equivalent to this situation.
(Sry, just realized someone posted about this already

pls move on and ignore me xD )