Let $s \propto (1, \ldots, 1)$ be a d-dimensional unit vector, so that $s_i = 1 / \sqrt{d}$. Let ${\cal V} = \{ u \in {\mathbb S}^{d-1}:u \cdot s = \cos \theta \}$ be a collection of unit vectors that form an acute angle with $s$. Suppose that $\cos \theta \rightarrow 1$ as $d \rightarrow \infty$ at a rate to be specified below.
I am interested in the following: what fraction of elements of ${\cal V}$ have the same sign as $s$? Put another way, suppose that $U$ is uniformly distributed over ${\cal V}$, then what is ${\mathbb P}[\cap_{i=1}^d\{{\rm sign}(s_i) = {\rm sign}(U_i)\}]$?
It is clear that when $d(1-\cos^2\theta) \rightarrow 0$ that all elements of ${\cal V}$ have the positive sign. However, I am not sure how to obtain a closed form solution when $d(1-\cos^2\theta) \rightarrow \infty$?