Which of the doors is the correct one changes per game. However, it is always the same door for all players in the room, so keeping an eye on where other players leave can help save precious time. If a new player enters the minigame, the positions of doors will change for all players, possibly to an already opened door, however the last door will always work despite this.
If a player tries to pick the doors in the final room, they will instead cause the player to exit the minigame, akin to the exit doors.
The chance of successfully opening the doors decreases by 1/256 every room. The following are the rates in the first room:
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