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.
“A tiger, you say .”
((Meaning of life)(war))
Well, that’s one way of looking at it.