If we let $X_n$ be the number of heads after flipping a fair coin $n$ times, what is
$\lim \limits_{n \to \infty} \Bbb P(X_{2n} > n + \sqrt n, X_{4n} > X_{2n} + n - \sqrt n)$?
The hint is to use the convergence of a random walk to Brownian motion, so my initial thoughts were about relating the probabilities to the normal distribution, but I wasn't sure how that relates to the hint, and in any case the scenario doesn't seem to mirror a symmetric random walk, as $X_n$ is just counting the number of heads.