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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 $n$?

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You should look instead at the probability $p_m$ that $$ y\notin \bigcup_{i=1}^m \mathcal H(a_i) \;, $$ or, in other words, that $\langle y, a_i \rangle < 0$ for all $a_i$. Since $a_i$ are independent, $$ p_m = \left( \mathbf P \{ \langle y, a \rangle < 0 \} \right)^m \;, $$ where $a$ is uniformly distributed on the half-sphere determined by $x$. Finally, $\mathbf P \{ \langle y, a \rangle < 0 \}$ is just the ratio of the volume of the intersection of the half-spheres determined by $x$ and $-y$ and the volume of the unit radius half-sphere in $\mathbb R^n$.

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  • $\begingroup$ Thank you for your answer. Is the ratio just $\alpha / \pi$? I wonder whether the ratio depends on the dimensionality n? $\endgroup$
    – Cong Ma
    Apr 15, 2017 at 18:32
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    $\begingroup$ Yes, indeed it's just $\alpha/\pi$ (which can be seen by looking at the plane spanned by $x$ and $y$ and its orthogonal complement). $\endgroup$
    – R W
    Apr 15, 2017 at 18:50
  • $\begingroup$ @R W, if you have time, could you please have a look at my other question on random spherical caps? Thank you very much! Any hints would be appreciated. $\endgroup$
    – Cong Ma
    Apr 23, 2017 at 19:23

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