Any mathematical relation between these conditional probabilities

Question

Let's say $A$, $B$ and $C$ are three different events. Is there any mathematical relation between these conditional probabilities: $\Pr(A\mid B,C)$, $\Pr(A\mid B)$ and $\Pr(A\mid C)$?

Note: In the relation there could be expressions such as $\Pr(A)$, $\Pr(B)$, $\Pr(C)$, $\Pr(B\mid C)$, etc. but not conditional probabilities in which $A$ is the condition, e.g., $\Pr(B\mid A)$. In other words, we only observe the events $B$ and $C$, and we would like to estimate the probability distribution of the event $A$.

Answer 1

As an example of the issue, consider $B$ and $C$ independently having probability $\frac12$. Then consider $A$ as $B \text{ XOR } C$, i.e. $A$ occurs if one but not both of $B$ and $C$ occur. Now look at the conditional probabilities:

• $\Pr(A\mid B)=\frac12$
• $\Pr(A\mid C)=\frac12$
• $\Pr(A\mid B,C)=0$

In general it is possible for $\Pr(A\mid B,C)$ to take any value (including $0$ and $1$) given that $\Pr(A\mid B)$ and $\Pr(A\mid C)$ have any values (except perhaps $0$ and $1$).