= Solution
Let $\tau_{\mathbb D}$ be the first hit of the closed unit disc. The conformal map $z\mapsto1/z$ sends its exterior to the punctured unit disc and sends $iy$ to $-i/y$. By conformal invariance, the hitting distribution on the unit circle is harmonic measure viewed from $-i/y$. As $y\to\infty$, this point tends to zero, where harmonic measure is normalized arc length by rotational invariance. Hence for every Borel $E\subset\partial\mathbb D$,
$$
\lim_{y\to\infty}
\mathbb P^{iy}(B_{\tau_{\mathbb D}}\in E)
=\frac{\operatorname{length}(E)}{2\pi}.
$$
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