Please consider the case where I have 'N' rods of length L (and width W) placed on a one- or two-dimensional surface with dimensions [0, A] in 1D, and [ [0, A], [0, B] ] in 2D.  For the two-dimensional case, L and W are << A or B.  

As a function of the number of rods N, and the relative dimensions of the rods and the surface on which they are placed, is there a reasonably easy derivation for the number of expected intersections between rods / the probability of an intersection occurring?  I feel like this should have been solved somewhere in the literature, but I was unable to find anything.