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 $M$.

  • $\begingroup$ This is nothing but the probability for the simple symmetric random walk on $\mathbb{Z}$ with absorbing barriers at $m$ and $(-m)$, not being absorbed before time $N$. You may look up for that. $\endgroup$
    – MikeG
    Feb 3, 2022 at 4:56


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