Random Walk on n+1 cycle, T is time such that walk returns to the initial vertex. Find the probability of visiting every vertex prior to time T.
Where walk moves clockwise w.p. $p \in (0,1)$ and counter-clockwise w.p. $(1-p)$.
This seems to be similar to gamblers ruin or any random walk with boundaries.
My approach is to consider : $S_0 = k$ where $k = 0,1,\ldots,n$
It would move to $k-1$ with probability $1-p$ so then consider the probability of hitting $k+1$ before hitting $k$
and similarly, walk moves to $k+1$ with probability $p$ then consider probability of hitting $k-1$ before $k$.
Im not sure if this is the correct approach, it seems along the correct lines but not quite.
I think this reduces to the gamblers ruin problem. Let me know if you think this sketch is reasonable -- I'd be happy to provide more detail.
Let's say we start at node $0$. We either move to $1$ or $n$ in the first step. Let's consider the case we move to $1$, since the other one is similar.
We are now asking about the probability that a biased random walker on $\mathbb{Z}$ that starts at $1$ reaches $n$ before it reaches $0$; the main point is that if we start the walk with $X_0 = 0, X_1 = 1$, then reaching $n$ before hitting $0$ again is equivalent to touching every node before reaching $0$ again.
Here the bias is that the random walker moves up with probability $p$ (a clockwise motion) and down with probability $1-p$ (counter clockwise motion). (For the other case the bias is swapped.)
The biased gamblers ruin is analyzed here: https://en.wikipedia.org/wiki/Gambler%27s_ruin#Unfair_coin_flipping