Probability - Independence Question

[jmcgraw]

I'm trying to get the idea of independent events well grounded in my mind, but I'm having some difficulty.

First of all, what would a venn diagram for independent events look like? I know you have the two events intersecting, and the intersection is equal to the product of the probabilities of the two events... But all I can picture is a normal diagram of two events intersecting, which by eyeballing looks just like the diagram of any 2 events (depndent or independent). Is there something unique to the diagram of independent events?

Mathematically, I just need to know that $$P(A \cap B) = P(A)P(B)$$ . But I can't form a graphical intuition about this, which I think would help me a lot.

Incidentally, I can't help but keep picturing the venn diagram of mutually exclusive events as a representation of independent events... But I know that is dead wrong, since mutually exclusive events are dependent.

Can anyone help me clarify this in my mind? Thanks!

 

[arildno]

If B is independent of A, it means that the intersection of B with A is the SAME fraction of A's total area as the intersection of B with not-A is of not-A's area.

 

[jmcgraw]

If B is independent of A, it means that the intersection of B with A is the SAME fraction of A's total area as the intersection of B with not-A is of not-A's area.

I think I just got more confused.

Isn't the intersection of B with not-A the same as not-A? giving the fraction to be simply 1?

Obviously I must be missing something, big time.