+Bas Spitters, very interesting: all three definitions conform to the intuition that a process is random if it defeats every attempt at simplifying its description!

+Phil Stracchino _Even though we cannot predict the random decay of any given atom, we can accurately predict the behavior of the sample as a whole_

is precisely the crux of the matter. That a process is hopelessly complex does not mean we cannot say anything at all about it: we can at least distinguish it from other complex processes, which is in effect what probability theory does for us. Deterministic examples you may have had in mind include Brownian motion as well as something as "simple" as a coin toss. The latter is a very clear illustration of what I'm trying to say: the process whereby the coin tumbles through the air before eventually settling on one of its faces, is hopelessly complex, but the

*symmetry* of the coin allows us to predict that it will come up tails as often as heads if we toss it toss it sufficiently many times.

It is true that there is a difference between classical (deterministic) and quantum randomness: in the former case, the complexity involved is evident (trillions of particles bumping into each other, a coin tumbling through the air), whereas in the latter randomness is part of the axiomatic framework. We simply refuse to explain

*why* quantum processes seem random, we just accept that they are (well, unless you count the Feynman path integral interpretation, which makes quantum processes look like a kind of Brownian motion, but that's just convenient computational framework, rather than anything actually observed).

Finally, a very important point is that mathematical definitions of randomness in

+Bas Spitters's link describe some idealized,

*absolutely irreducible* complexity. However, when we apply such idealized models to real world phenomena, they are always mere approximations. Which is why I spoke about

*intractable* complexity in the original post: when we encounter an intractably (for us, at the present stage of development!) complex phenomenon, we model it as random. But we may yet develop to a point where what we see as random now will not seem random at all.