I have always found the notion of randomness hard to understand. It is not that I don't understand what people are saying when they use the word "random" in certain everyday contexts, such as when someone says "I was randomly assigned to this team", but that I don't have a clear understanding of how randomness is supposed to be understood as related to probability, chance, and predictability.
Right after the first appearance of the word "random" in his An Introduction to Probability and Inductive Logic, Ian Hacking remarks, "Randomness is a very hard idea." This really struck a chord with me when I first read it. Although Hacking goes on to give something like a definition of "randomness", he is very careful in formulating it:
Outcomes from a chance setup are random (we think) if outcomes are not influenced by the outcomes of previous trials.
Notice the parenthetical "we think".
I am going to read with some colleagues Hacking's famous book The Emergence of Probability. Hope this will help me understand randomness better.