Markov Chains & Long-Run Behavior · Challenge problem

A fair random walk on $\{0, 1, 2, 3, 4, 5\}$ moves $\pm1$ with probability $1/2$ each and is absorbed at $0$ and at $5$. Starting at $2$, what is the expected number of times the walk is at state $3$ before absorption? Use the fundamental matrix $N = (I - Q)^{-1}$. Express your answer as a common fraction.