While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over something.

I am currently thinking of

(1) Linear independence in linear algebra, for elements of a $R$-module,

(2) Algebraic independence for elements of a $K$-algebra (over a field $K$), on which there is a natural action of $K[x_1,\dots,x_n]$,

(3) Free groups, presentation by generators and relations.

(4) Independence in probability.

(5) Shelah's notion of abstract independence for types in model theory, particularly in stable theories.

Q1. Any other natural/interesting examples ? Or comments about these ones ?

Whereas I clearly see a connexion between the first 3 items : the question of the existence of a non trivial 'equation' of a certain kind bounding the elements, item (4) seems of a different nature (at least I do not understand its nature).

Q2. Are there analogies or more between these items ? Is there someting that the probabilistic notion of independence shares with say item (1) ? What is a good notion of independance and why ? Why is it usefull ?

These questions are very naïve, so naïve answers are allowed !