Let us consider a random walk denoted by $S_n$ and let $M_n$ be the maximums of the random walk. Now let us consider that this random walk will end at some point $k$. So I am stuck how to prove this equality:
$$P(M_n \geq r , S_n=k) = P(S_n=2r-k)$$
I can prove the right-hand side but someone please solve the left-hand side.
There is a bijection between walks which satisfy $S_n=k,M_n=r$ and walks which satisfy $S_n=2r-k$. To see that, reflect the path from the first time the walk reaches $r$ about $r$, so that from that point on +1 becomes -1 and vice versa.