Markov Chains & Long-Run Behavior · Challenge problem

A mouse wanders through three rooms. Room 1 has three doors: to room 2, to room 3, and to the cheese. Room 2 has two doors: to room 1 and to room 3. Room 3 has four doors: to room 1, to room 2, and two doors to the cat. Each step the mouse leaves by a uniformly chosen door of its current room; the cheese and the cat are absorbing. Starting in room 2, what is the probability that the mouse reaches the cheese? Express your answer as a common fraction.