I understand that mutually exclusive events have no intersection, and this can be shown on a Venn diagram as separate sets. And I understand that independent variables are such that $P(AB) = P(A)P(B)$, so if $P(AB) = 0$ then $P(A) = 0$ or $P(B) = 0$. But can you show independent variables on a Venn diagram? Why is it that $P(AB)$ isn't always zero, if A and B are independent?

Consider flipping a single coin. Then $P(heads) = 1/2$ and $P(tails) = 1/2$. However, $P(HT) = 0$, since it is impossible to show heads and tails simultaneously. So I would conclude that H and T are not independent in this case. And would you say that H and T are mutually exclusive?

Now consider flipping two separate coins, again with $P(H) = P(T) = 1/2$. This time $P(HT) = 1/4$, as the coins are independent. They do not influence each other. But are the coins mutually exclusive? It seems they can't be, since $P(HT)$ is not zero.

I'm getting confused about mutually exclusive vs. independence. Please provide examples of mutually exclusive and independent, mutually exclusive and dependent, not mutually exclusive and independent, and not mutually exclusive and dependent.