The notions of algebraic independence (as in linear independence) and probabilistic independence are both captured in the notions of an independent subset of a Boolean algebra and the notion of an independent family of partitions.
Suppose that $B$ is a Boolean algebra. Then a subset $R\subseteq B$ is said to be independent if $R$ freely generates a subalgebra of $B$. Then $R$ is independent if and only if whenever $r_{1},...,r_{n}\in R$ are distinct elements and for all $i$ either $s_{i}=r_{i}$ or $s_{i}=r_{i}'$ we have $s_{1}\wedge...\wedge s_{n}\neq 0$. Recall that a subset $p$ of a Boolean algebra $B$ is a partition if $0\not\in p$, $\bigvee p=1$ and where $a\wedge b=0$ whenever $a,b\in p,a\neq b$. A collection of partitions $\mathcal{P}$ of $B$ is said to be independent if whenever $p_{1},...,p_{n}\in\mathcal{P}$ and $a_{1}\in p_{1},...,a_{n}\in p_{n}$ then $a_{1}\wedge...\wedge a_{n}\neq 0$.
This notion of independence in a Boolean algebra can be thought of as both an algebraic independence and a probabilistic independence. For example, if $R_{1},...,R_{n}$ are events in a probability space with $P(R_{i})\in(0,1)$ for $1\leq i\leq n$ and $R_{1},...,R_{n}$ are independent in the probabilistic sense, then $R_{1},...,R_{n}$ are independent in a Boolean algebraic sense.
The main result about independence in Boolean algebras is the result by Balcar and Franek which states that every infinite complete Boolean algebra $B$ has an independent subset $R\subseteq B$ with $|R|\subseteq|B|$ and hence a free subalgebra $A\subseteq B$ with $|A|=|B|$. One corollary of this result is that if $A,B$ are complete Boolean algebras with $|A|\leq|B|$, then there is a surjective Boolean algebra homomorphism $\phi:B\rightarrow A$. Furthermore, Balcar and Franek go on to show that complete Boolean algebras have not just large free algebras, but they also have large independent sets of large partitions in the following sense.
Suppose that $B$ is a Boolean algebra. Recall that the saturation $Sat(B)$ of $B$ is the least cardinal such that every partition of $B$ has cardinality less than $Sat(B)$. Then $B$ is said to be well-semifree if there is an independent family $\mathcal{P}$ of partitions such that if $\lambda<Sat(B)$, then $\{p\in\mathcal{P}:|p|=\lambda\}=Sat(B)$. A Boolean algebra $B$ is said to be saturation homogeneous if whenever $B\simeq A\times C$ and $|A|>1$ then $Sat(A)=Sat(B)$.
$\mathbf{Theorem}$(Balcar and Franek) Every saturation homogeneous
complete Boolean algebra is well-semifree.
The condition that the complete Boolean algebra $B$ is saturation homogeneous a very minor restriction since every complete Boolean algebra is a direct product of saturation homogeneous complete Boolean algebras.
