TITLE: Probability of A given B given C
QUESTION [5 upvotes]: How would I find P ((A|B)|C)? Do I substitute the formula for P (A|B), P=(A ∩ B))/(P (B)), and then redo the function to get (P (A ∩ B ∩ C))/(P (C)^2)?
I heard that P=(A|B ∩ C) can be used, but why would that be equivalent, unless you assume that C is independent of A?  Given a three-circle Venn diagram, the initial sample size would be too restricted.

REPLY [5 votes]: The notation $\mathsf P((A\mid B)\mid C)$ is not standard.   There should only be one bar between the event being measured and the condition.   When conditioning over two events, take the conjunction.
Both $\mathsf P(A\mid B, C)$ and $\mathsf P(A\mid B\cap C)$ mean the conditional probability of $A$ given $B$ and $C$.
$$\mathsf P(A\mid B\cap C) = \frac{\mathsf P(A\cap B\mid C)}{\mathsf P(B\mid C)} = \frac {\mathsf P(A\cap B \cap C)}{\mathsf P(B\cap C)}$$