[mturmon]

"...dealing with genuinely random phenomena is rare. More common is that our knowledge is limited."

Quantum mechanics is the best thing we have to an exactly correct physical theory, and it is tied completely up with probability. So probability in the form of "genuinely random" is very fundamental.

As you go up from that level, our knowledge is limited at every turn, so it's hard to separate the one from the other ("genuinely random" vs. limited knowledge).

For example, you may have a large-scale physical system governed by differential equations, but the boundary conditions are not exactly known or knowable, or the governing equations are not closed (i.e., they depend on other things outside the system).