Imagine two events $A$ and $B$ that are not mutually exclusive, such that $P(A) = 0.3$ and $P(B)=0.4.$ Consider the Venn diagram of the two overlapping sets, and visualize moving them closer together or further apart, thus varying the size of the overlapping region $A \cap B$. It should be $\color{darkorange}{\text{clear}}$ that $P(A \cap B)$ could take on any value between $0$ and $0.3$, but of that infinite set of possible values, the only one that would make $A$ and $B$ independent would be $P(A \cap B) = 0.12.$
After reading this answer (excerpted above), I drew Venn diagrams below for $P(A \cap B) =0$ (when $A$ and $B$ are disjoint), $P(A \cap B) =0.12,$ and for $P(A \cap B) =0.3$ (when $B \subsetneq A)$.
However, it is $\color{darkorange}{\text{not
clear}}$ to me that the only one that would make $A$ and $B$ independent would be $P(A \cap B) = 0.12$. I still can't intuit or visualize independence.
Best Answer
It's not feasible to represent independence. Most people think drawing Venn Diagrams that are disjoint demonstrates independence, but that is false. If two events are disjoint, they are inherently dependent. Because if I know A occurs, I know B cannot occur and vice versa.
So what about the opposite: drawing Venn Diagrams that intersect? This method isn't quite useful for visualizing independence either. For example, let A be the event Susan studies for her test and let B be the event Susan eats chocolate. The intersection in the Venn Diagram means that Susan studies for her test and she eats chocolate. Let's say these two events are independent. That doesn't mean these two events can't co-occur, i.e. intersect. They can. Likewise, let's say the events are dependent. They can still co-occur or not co-occur. So there, we have shown that we can't demonstrate independence via intersection either.