Which door I think the princes chose

In the short story, the Lady of the Tiger, it tells about a Barbarian king being the ruler of a court/arena. In the arena, there are two, sturdy, soundproof doors that contain a lady and a tiger. The person in the arena can only choose one door, one with a lady he must marry or a deadly tiger.
The book says that the person in the arena loved the king’s daughter, so he put a person his daughter hated. Later, the princes pointed at one of the doors and the convicted person opened it. I think that the princes just randomly chose one because princes already lost her lover. If the princes picked the one with the lady, she wouldn’t be able to marry the youth. That is because the youth would spend all his time with the lady, which he has to marry. Even, is the princes managed to get to the youth, she would still have to face the lady, whom the princes hate. If she pointed at the door with the tiger, the youth would be dead.
Either way, the princes would not be able to spend time with her lover, because he is would be either dead or living with the lady. This is why I think that the princes just randomly chose one of the doors.