Are these conditional probabilities related?

Question

Knowing $P(A|B)$ & $P(A|C)$, would it be possible to infer $P(A|B,C)$?
I looked at this other question: Any mathematical relation between these conditional probabilities but it doesn't solve my problem as he only uses mutually exclusive probabilities.

Answer 1

Just look at simple examples.

Suppose we are tossing a fair penny and a fair dime. Let $A$ be the event "the penny comes up $H$". Let $B$ be the event "the dime comes up $H$" and let $C_1$ be the event "the two coins do not match". Then $$P(A|B)=\frac 12\quad \&\quad P(A|C_1)=\frac 12\quad \&\quad P(A\,|\,(B\cap C_1))=0$$

Now let $C_2$ be the event "the coins do match". then

$$P(A|B)=\frac 12\quad \&\quad P(A|C_2)=\frac 12\quad \&\quad P(A\,|\,(B\cap C_2))=1$$

Answer 2

No. For example, let $X$ be a roll of a standard die, and let $B$ be the event $X<4$ and $C$ be the event that $X$ is even.

Now if $A$ is the event $X\equiv 1\pmod 3$ then $P(A)=P(A\mid B)=P(A\mid C)=1/3$, but $P(A\mid B,C)=0$.

However, if $A$ is the event $X\equiv 2\pmod 3$ then again $P(A)=P(A\mid B)=P(A\mid C)=1/3$ but this time $P(A\mid B,C)=1$.