Let $\omega_1, \omega_2, \ldots$ be uniform iid on $\{-1,1\}$, and let $X_n = \sum_{i=0}^n \omega_i$ be the corresponding simple random walk. Fix some integer $N$, and let $h^+_N$ be the first time $X_n$ hits $N$, $h^-_N$ the hitting time of $-N$, $c^-_N$ the hitting time of $-(N-1)$, and finally let $e^+_N$ be the hitting time of $N+1$. Let $Q$ be the event that $h^+_N < h^-_N$ and $c^-_N < e^+_N$.

**What is $\mathbb{P}(Q)$?** I don't necessarily need an exact formula, just to know how it behaves as $N$ grows big.

The picture here is that if $Q$ occurs, then the random walk either first hits $N$ before bounding back to hit $-(N-1)$, or it very nearly hits $-N$ before bounding back to hit $N$. Either way it makes a "big swing". You could also think of the event $Q$ as containing (half of) the boundary between hitting $N$ first and hitting $-N$ first, inside the big set $\{-1,1\}^\infty$.