there is a sample space of lines, one on which the affine group of the plane acts. a probability measure is sought on this space, invariant under the symmetry group. if, as in this case, we can find a unique such invariant measure, then that solves the problem of formulating accurately what 'random line' means and expectations become integrals with respect to that measure. (note for example that the phrase 'random chord of a circle' can be used to construct some paradoxes--for example bertrand's paradox.)