Is knowing $P(A), P(A,B)$, and $P(A,C)$ sufficient to find $P(A \mid B,C)$?

Question

$A,B,C$ are possible values for the random variable $A', B', C'$ respectively. My intuition says no because you don't know if $B$ and $C$ are independent or not, and by Bayes' rule $P(A \mid B,C) = \frac{P(A, B, C)}{P(A, C)}$.

Is it possible to find $P(A , B , C)$ and $P(A , C)$ with just the information above?

Answer 1

In general, no. Let $X_1, X_2, X_3$ be iid Bernoulli $1/2$ r.v. s, and let $Y_1, Y_2$ be iid Bernoulli 1/2 random variables, and let $Y_3 = Y_1 \oplus Y_2$ (XOR addition)

Here, you can check that all the pairwise pmf s of $X$s and $Y$s are the same, but their mutual pmf is different.