Expected time for winning in biased Gambler's Ruin

Expected time for winning in biased Gambler's Ruin

Consider the random walk $X_0, X_1, X_2, \ldots$ on state space
$S=\{0,1,\ldots,n\}$ with absorbing states $A=\{0,n\}$, and with
$P(i,i+1)=p$ and $P(i,i-1)=q$ for all $i \in S \setminus A$, where $p+q=1$
and $p,q>0$.
Let $T$ denote the number of steps until the walk is absorbed in either
$0$ or $n$.
Let $\mathbb{E}_k(\cdot) := \mathbb\{\cdot | X_0 = k\}$ denote the
expectation conditioned on starting in state $k \in S \setminus A$.
How to compute $\mathbb{E}_k(T|X_T = n)$?