Basic probability - independent events and the empty set I'm having a problem understanding this seemingly trivial aspect in basic probability theory.
Suppose there is two independent events, $A$ and $B$.
Their probabilities are known and they are $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{4}$.
For independent events, $$P(A \cap B) = P(A) \times P(B)$$ and so we have $$P(A \cap B) = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12}$$
But if I plot a Venn diagram for this state space, I see that $A \cap B = \emptyset$ and, as such, $P(A \cap B) = 0$ and certainly not $\frac{1}{12}$.
Where have I gone wrong?

 A: (This is a comment, not an answer, but I can't put an image in a comment.)
A better picture to have in mind for independence is something like:

A: Let's actually consider two different events.
Let P(A) be the probability that Ann will come to your party.
Let P(B) be the probability that Bill will come to your parry.
Now, since P(A) and P(B) are independent, Ann showing up has no impact on whether or not Bill will show up, and Bill showing up has no impact on whether or not Ann will show up.
So we end up 4 possibilities:
Ann and Bill show up.
Ann shows up but Bill doesn't.
Bill shows up but Ann doesn't.
Neither Bill nor Ann show up.
Now we are interested in the probability of both of them showing up.
There is a 1/3 probability that Ann will show up.  Let's assume for a second that Ann does show up.  The probability that Bill will also show up is 1/4 (since it is not affected by Ann showing up).
So of that 1/3 chance of Ann showing up, we have a 1/4 chance of Bill showing up.
In other words we have a 1/4 chance of a 1/3 chance.
$\dfrac{1}{3}\times\dfrac{1}{4}=P(A)\times P(B)$.
The diagram drawn by Gregg Martin illustrates this very well.
For two probabilities to be disjoint as you have in your diagram would actually imply that they are dependent (e.g. have an impact on each other).
Your diagram might be better for an example like:
A drug is tested and it is discovered that it has 1/3 chance of working and a 1/4 probability of making it worse.   The white part would be the probability that it has no effect on the problem it is supposed to solve.  And, if it works, it can't also make the problem worse, thus P(A) and P(B) are completely dependent (and appear just as in your diagram).
