The tiger always wins.
Not if the designated selector selects the door with the lady behind it, assuming that being selected is the preferable outcome.
I'd say the tiger only wins approximately 0.50000000/1.0000000 of the time, and that's if said feline keeps reasonably quiet while ensconced behind the closed door (and the lady doesn't do a bang-up tiger impression) and the door has enough weather stripping to hold back the odor of tiger (which I found to be very distinctive and intense during my brief encounters with them).
Perhaps I'm picking a nit.