Surely the Cauchy distribution you're replying to doesn't have any steradians going on, right? It's a 1D distribution and the gun described would point towards a random point on a circle, not on a sphere.
I was actually thinking of a separate example from physics (years ago) calculating the classical flow of particles through an aperture. Where you assume particles have maxwell-Boltzmann distribution of speeds and consider them travelling in all directions over 3D and also the effective size of the aperture vs the azimuthal angle of travel. It's quite similar to this example actually but given the speeds you also must account for which velocity range [v,v+dv] as a function of angle will put a particle through the hole in time dt.
