After studying statistics for a reasonable amount of time, my personal view of randomness is the distribution of some metric conditioning on a set of information. The randomness is a function of how much we know about this particular instance of the metric.

Let's take the classic six sized die example. If I'm about to roll it, one could make an argument that given all the forces acting on that die, the outcome is deterministic. We can imagine that if we understood and could perfectly control all these forces, we could make a machine that always gives us the roll we want. It's just a lot of physics.

But those are a lot of forces I don't understand. So I'm just going to condition on the fact that the die is fair (I assume) and average over all those unknown forces. In that case, we can think of an near infinite set of possible forces acting on this die, and we believe in 1/6 cases, a 1 shows up. Thus, if all I know is that the next roll is going to be from a fair die but I don't know all the acting physics on that die, the best I can do is say that the outcome belongs to a set in which 1/6 of the outcomes are 1, so I say to the best of my knowledge, P(roll = 1) = 1/6. I'm essentially averaging over this unknown forces that would have made my metric deterministic had I known them.

As another example, suppose some asks "what's the distribution of heights of 23 year old men?". To be honest, I'm not exactly sure the answer to this, but if someone told me that it was approximately normally distributed with mean 5'8" and standard deviation 4" that would seem believable to me. But note that if I provided you with more information, i.e., "what's the distribution of heights of 23 year old men who's father was 6'1"?", I've provided you with more information that allows you to narrow down the larger set (23 year old men) to a smaller subset (23 year old men with fathers 6'1"). Mostly likely there is less variability in this subset which then reduces the variability of the metric in this subset. Thus we see that the variability of a particular metric (height) is very dependent on exactly what we are conditioning on having known (age? gender? height of father?).