Section I (Question 29): A trained mouse lives in the house…

Section I (Question 29): A trained mouse lives in the house shown below.    A bell rings at 9 AM every day. When the bell rings, the mouse is trained to choose a door in the room it is in. When it chooses doors, it is equally likely to choose any of the doors in the room it is in. Given that the mouse is currently in room A, how many days does it take on average in the long run for the mouse to escape from the house? Show your work.