Classify the states of a markov chain a)
P =$\begin{bmatrix}
{1-2p} & 2p & {0} \cr
{p} & {1-2p} & {p} \cr
{0} & 2p & {1-2p} \cr
\end{bmatrix}$
b)
P = $\begin{bmatrix}
0 & p & 0 & 1-p \cr
1-p & 0 & p & 0 \cr
0 & 1-p & 0 & p \cr
p & 0 & 1-p & 0 \cr
\end{bmatrix}$
In each case, calculate $p_n(i, j)$ and the mean recurrence times of the states (For each
state j we may dene a corresponding stopping, the rst return time, as $T_j = min(n \ge 1:  X_n = j)$. Then the mean recurrence time, j , of j is given by: j = $E(T_j|X_0 = j)$.
Im not sure how to approach this question since, it would be different for different values of p.
Any help would be appreciated,
thanks
Ok so for p = 1/2 ive established that $s_0$ and $s_2$ are transient and $s_1$ is recurrent, but how could i show this for p other than 1/2?

This is called Kolmogorov forward equations that are used to find stationary probabilities: $\pi_k=\lim_{t \to \infty} (P(X(t)=k)$:
$$
\pi_1=(1-2p) \pi_1 + p \pi_2\\
\pi_2=2p \pi_1 + (1-2p) \pi_2 + 2p \pi_3
$$
You do not need the 3rd equation since you have a standardizing condition: $\pi_1 + \pi_2 + \pi_3 =1$. Same approach for the 2nd MC. Can you handle it from here?
EDIT: 
In this case you have a system of linear equations $\mathbf{A} \mathbf{x} = \mathbf{b}$, which you have to solve for $\mathbf{x} = [\pi_1, \pi_2, \pi_3]$, where $\mathbf{A} =[-2p, p,0; 2p , -2p, p;1,1,1]$, $\mathbf{b}=[0,0,1]$
