Let $(X_n)_{n \geq 0}$ be a markov chain on $I={1,...,12)$, with stochastic matrix:
$$p_{i,j}=\begin{cases} p & if \ j=i+1 \ mod \ 12 \\ q=(1-p) & if \ j=i-1 \ mod \ 12 \\ 0 & o/w \end{cases}$$, where $p\in(0,1)$.
Suppose we begin with $X_0 = 6$, then what is the probability that the chain hits state 11 in finite time?
I understand that this means we start in state 6 and we can move to adjacent numbers (i.e. in $X_1$ we will be in state 5 or 7).
However, how would I attempt to calculate the hitting probability, $h_i^A$?
If you draw the Markov chain, it is a ring with $12$ nodes.
Every finite closed communicating class is recurrent.
Hence the probability of hitting state $11$ in finite time is $1$. The probability of hitting any state in finite time is $1$.