On a finite connected graph if a random walked is choosing the next vertex uniformly at random from among the edges of its current vertex, then it looks quite obvious to me that given an infinite walk it can go from any vertex to another with probability $1$. One could very well use this property of such a random walker to define connectedness of a finite graph.
BUT, does there exist (or is there a need?) to come up with anymore formal a proof of this observation? (...it looks quite tautological to me!..)
It is quite straightforward to prove formally: Let $d$ be the diameter of the graph, the length of the longest shortest path between two vertices. Let $p = \frac{1}{n}$, where $n$ is the number of vertices. Let $v$ be an arbitrary vertex. Then at each time point, there is at least a $p^d$ chance of reaching $v$ in $\leq d$ steps (since there is a path from our current point to $v$ of length at most $d$). This means that the probability that $v$ is not reached in the first $kd$ steps is at most $(1-p^d)^{k}$. This probability converges to $0$ for $k \to \infty$.