how to calculate $P(C|A∩B)$ from $P(C)$ $P(C|A)$ and $P(C|B)$ where the universal set is $A\times B$

Question

I don't know if I'm using the right terminology so I'll try to clarify what I mean:

$U=\{(a',b')\mid a'\in A'\ and\ b'\in B'\}$

$a\in A$

$b\in B$

$A=\{(a, b')\mid b'\in B'\}$

$B=\{(a', b)\mid a'\in A'\}$

$C \subset U$

given $P(C|A)$ $P(C|B)$ and $P(C)$ can you solve for $P(C|A∩B)$, if so how, and if not why, and what extra information is needed.

Related questions:

How to Calculate the conditional probability $P(Z|AB)$ if we know $P(Z)$, $P(A)$, $P(B)$, $P(Z|A)$, $P(Z|B)$, $P(AB)$

Estimating conditional probability

Answer 1

Two scenarios can happen since $A\cap B = \{(a,b)\}$. Either $(a,b)\in C$ or not. In the second case your probability is $0$ by definition. In the first you get \begin{align*} P(C|A\cap B) &= P(A\cap B\cap C)/P(A\cap B)\\ &= P(\{(a,b)\})/ P(\{(a,b)\})\\ &= 1 \end{align*}

This is all true if and only if $P(A\cap B)>0$, in the case $P(A\cap B)=0$, then $P(C|A\cap B)=0$.