This is why I suggested Frank's paper. It talks about why random local processes - like tossing the needle - are indistinguishable from an aggregation of deterministic local processes.
You see the same story with fractals. The same patterns can be generated using random processes or deterministic iterative equations.
There is something important going here that could be related to physical theory - the existence of local symmetries (as in gauge symmetries of particle physics) within the self-organising contexts of global constraints (as in soliton models of particles).
But I don't see random numbers as being the key. Rather it is the deeper principles of symmetries and symmetry-breaking.