Let $S^{n-1}$ be the unit sphere in $n$ dimensional Euclidean space. Define the spherical cap at $x \in S^{n-1}$ with angle $\theta$ to be $C(x,\theta) = \{z \in S^{n-1} \mid z^\top x \geq \cos(\theta)\}$.

Let $x,y \in S^{n-1}$ be two unit vectors with angle $\alpha \in (\frac{\pi}{2}, \frac{3\pi}{4})$. Choose $m$ unit vectors $\{a_i\}_{i=1}^m$ independent and uniform over $C(x,\frac{\pi}{2})$, what is the probability that $C(y,\beta) \subseteq \bigcup_{i=1}^{m}C(a_i, \frac{\pi}{2})$ for some $\beta \in (0, \frac{\pi}{4})$?

This is similar to the classical sphere covering problem in geometric probability. The two differences here are that: first the vectors are chosen from $C(x,\frac{\pi}{2})$ rather than the whole sphere, second, it only need to cover a small spherical cap rather than the whole sphere.