How to get conditional probability from joint conditional probability?

Question

Given a joint conditional probability of A intersect B given C, is there a way to get the conditional probability of A given C? What I mean is having something like this:

$\mathbf P(A\cap B\mid C) = \mathbf P(A\mid C)\cdot(\textrm{...})$

beside the conditionally independence formula:

$\mathbf P(A\cap B\mid C) = \mathbf P(A\mid C)\cdot\mathbf P(B\mid C).$

Thanks.

Answer 1

I think it should be $$\mathbf P(A\cap B\mid C) =\frac{ \mathbf P(A\cap B\cap C)}{\mathbf P(C)}$$ since, $$\mathbf P(X\mid C) =\frac{ \mathbf P(X\cap C)}{\mathbf P(C)}$$ where $X=A\cap B$