A quantity associated to a probability measure space Let $(S,P)$ be a (finite) probability space. We associate to $(S,P)$ a quantity $n(S,P)$ as follows:
The probability of two randomly chosen events $A,B\subset S$ being independent is denoted by $n(S,P)$.
Is there a terminology for this quantity? Is it equivalent to some other well known terminology in probability theory? Can one generalize this concept to infinite sample spaces? (And a possible generalization to arbitrary measure spaces?)
What is this quantity for the  sample space associated to the  experiment of rolling two different colored dice (the standard probability space this experiment generates)?
 A: The probability -- say $p$ -- for the experiment of rolling two different colored dice is
$$\frac{162601421574468954588}{2^{2\times36}}\approx0.0344322.$$
Here it is assumed that the random sets $A$ and $B$ are selected independently and uniformly at random (from the power set of the set $[6]^2=\{1,\dots,6\}\times\{1,\dots,6\}$), so that $P(A=a,B=b)=1/2^{2\times36}$ for all $a\subseteq[6]^2$ and $b\subseteq[6]^2$.
So, $p$ is the probability that the random sets $A$ and $B$ are independent. That is, $p$ is the probability that $\dfrac{|A|}{36}\dfrac {|B|}{36}=\dfrac{|A\cap B|}{36}$, where $|\cdot|$ denotes the cardinality.

Indeed, for the set $[36]_0:=\{0,\dots,36\}$, let $T$ denote the set of all triples $(m,n,k)\in[36]_0^3$ such that
$\dfrac m{36}\dfrac n{36}=\dfrac k{36}$ or, equivalently, $mn=36k$.
Let $I$ denote the set of all pairs $(a,b)$ of subsets of the set $[36]_0$ such that $a$ and $b$ are independent. Then $$(a,b)\in I \iff (|a|,|b|,|a\cap b|)\in T.$$
For any given $(m,n,k)\in T$,
$$N_{m,n,k}:=\big|\{(a,b)\in I\colon (|a|,|b|,|a\cap b|)=(m,n,k)\}\big|=\binom{36}m\binom mk \binom{36-m}{n-k}
=\frac{36!}{k!(m-k)!(n-k)!(36-m-n+k)!}.$$
So, the probability that the random sets $A$ and $B$ are independent is
$$p=\frac1{2^{2\times36}}\sum_{(m,n,k)\in T}N_{m,n,k}
=\frac{162601421574468954588}{2^{2\times36}},$$
as claimed.

Here are details of the calculations, with Mathematica:


The value $p\approx0.034$ agrees with simulation:

