[ykcirT]

Totally unrelated to poker, but I know there are some people who are very good at probability here, so I
thought I would take a chance. Can anyone figure this out:

Assume you are in a room with 4 doors, each heading in a different direction. You have to leave the room, so you randomly decide on a door and go through it. Every subsequent room also has 4 doors, which also lead in different directions (assume each door is exactly N,S,E,W). In each room, you make a totally random choice as to which door to go through. What is the probability that you will return to the original room if you have to keep leaving the room you are in until you either 1) return to the original room, or 2) Get yourself far enough away that you cannot return in 2 consecutive room changes. This drawing might help:

x x x 3 x x x
x x 3 2 3 x x
x 3 2 1 2 3 x
3 2 1 S 1 2 3
x 3 2 1 2 3 x
x x 3 2 3 x x
x x x 3 x x x

In other words: start at S, and start moving around. If you enter a room #3, you will never get back.
What is the probability of returning to the start?

Thanks. Sorry this is so off topic,
Rich