Sampling uniformly distributed random points inside a spherical volume

I am looking to be able to generate a random uniform sample of particle locations that fall within a spherical volume.

The image below (courtesy of http://nojhan.free.fr/metah/) shows what I am looking for. This is a slice through the sphere, showing a uniform distribution of points:

This is what I am currently getting:

You can see that there is a cluster of points at the center due to the conversion between spherical and Cartesian coordinates.

The code I am using is:

def new_positions_spherical_coordinates(self):
   radius = numpy.random.uniform(0.0,1.0, (self.number_of_particles,1)) 
   theta = numpy.random.uniform(0.,1.,(self.number_of_particles,1))*pi
   phi = numpy.arccos(1-2*numpy.random.uniform(0.0,1.,(self.number_of_particles,1)))
   x = radius * numpy.sin( theta ) * numpy.cos( phi )
   y = radius * numpy.sin( theta ) * numpy.sin( phi )
   z = radius * numpy.cos( theta )
   return (x,y,z)

Below is some MATLAB code that supposedly creates a uniform spherical sample, which is similar to the equation given by http://nojhan.free.fr/metah. I just can't seem to decipher it or understand what they did.

function X = randsphere(m,n,r)

% This function returns an m by n array, X, in which 
% each of the m rows has the n Cartesian coordinates 
% of a random point uniformly-distributed over the 
% interior of an n-dimensional hypersphere with 
% radius r and center at the origin.  The function 
% 'randn' is initially used to generate m sets of n 
% random variables with independent multivariate 
% normal distribution, with mean 0 and variance 1.
% Then the incomplete gamma function