Toss a fair coin multiple times. Let the total number of heads up to and includes $k$-th toss be $H(k)$ and denote by $T(k)$ the number of tails. The game ends whenever $D(k):=|H(k)-T(k)|= m$, where $m$ is a fixed integer.
Question: Fix $N\in\mathbb{N}$, what is $\mathbb{P}\left(D(k)<m, \forall k\in[0,N]\right)$? (i.e. the probability of the game has not ended at $N$)
My attempt: Let $M(k)=\max_{0\leq i\leq k}D(k)$, i.e. the running maximum of $D(k)$. Then the question is equivalent to $\mathbb{P}\left(M(N)<m\right)$. However, I'm not sure how to proceed on discovering the distributional properties of $M$.
