Consider a Markov chain $\left(X_{n}\right)_{n \geqslant 0}$ on states $\{0,1, \ldots, r\}$ with transition matrix $\left(P_{i j}\right)$, where $P_{0,0}=1=P_{r, r}$, so that 0 and $r$ are absorbing states. Let

$$A=\left(X_{n}=0, \text { for some } n \geqslant 0\right) \text {, }$$

be the event that the chain is absorbed in 0 . Assume that $h_{i}=\mathbb{P}\left(A \mid X_{0}=i\right)>0$ for $1 \leqslant i<r$.

Show carefully that, conditional on the event $A,\left(X_{n}\right)_{n \geqslant 0}$ is a Markov chain and determine its transition matrix.

Now consider the case where $P_{i, i+1}=\frac{1}{2}=P_{i, i-1}$, for $1 \leqslant i<r$. Suppose that $X_{0}=i, 1 \leqslant i<r$, and that the event $A$ occurs; calculate the expected number of transitions until the chain is first in the state 0 .