I distribute $N$ points on a sphere according to
$\displaystyle \theta=\arccos(2u-1) \\ \phi = 2\pi v $
where $\theta$ is the zonal angle, $\phi$ is the azimuthal angle, and $u$ and $v$ are random variables ranging uniformly from 0 to 1. All this to say, I am distributing so that points will not be more highly concentrated at the $\theta$ poles than at the equator.
Now, I do this for $N$ points. Given some angular separation $\Omega$, what is the probability that there will be some pair of points separated from each other by less than $\Omega$?
(Please don't ask me to show my work. I don't even know where to begin!)