conditional probability - conditioned twice Let $p(\cdot)$ be a discreet probability function and $A,B,C$ be events. What does $p(A|B|C)$ mean? Is this the same as $p(A|B,C)$ or is it:
if we treat $D=A|B$ as another event, $p(D|C) = \frac{p(D\cap C)}{p(C)}$?
How would $p(D \cap C)$ be computed then with $D$ defined earlier?

I think $$\frac{P\left(A\cap B\mid C\right)}{P\left(B\mid C\right)}=\frac{P\left(A\cap B\cap C\right)}{P\left(C\right)}/\frac{P\left(B\cap C\right)}{P\left(C\right)}=\frac{P\left(A\cap B\cap C\right)}{P\left(B\cap C\right)}=P\left(A\mid B\cap C\right)$$
has the best 'chances' here :).
