In the case of a simple random walk $\{S_n, n \ge 0\}$ what is $S_n$. I see this for $P\{S_n=i |\ |S_n|=i_{n-1},...,|S_1|=i_1 \} = \frac{p^i}{p^i+q^i}$
What does this mean? is it: the probability of going to state i after n steps (not sure about that) is $\frac{p^i}{p^i+q^i}$ , which is independent of n.
A random walk, in the context of Markov chains, is often defined as $S_n = \sum_{k=1}^n X_k$ where $X_i$'s are usually independent identically distributed random variables. My understanding of your given statement is the probability of the summation $S_n$ reaching value $i$ given all its previous history.