Let $x, y \in \mathbb{R}^{n}$ be two fixed unit vectors with angle $\alpha \in (\frac{\pi}{2}, \frac{3\pi}{4})$. Define the positive half space associated with a vector $z$ to be $\mathcal{H}(z) = \{h : z^\top h \geq 0\}$.

Choose $m$ unit vectors $\{a_i\}_{i=1}^{m}$ uniform over the set $\mathcal{H}(x) \cap \{h : \|h\|_2 = 1\}$, what is the probability that $y \in \bigcup_{i=1}^m \mathcal{H}(a_i)$?

How does this probability depend on $\alpha, m$ and $n$?