Multiconditional probability

Question

How can I calculate $P(X|A,B,C)$ knowing $P(X)$, $P(X | A)$, $P(X | B)$ and $P(X | C)$ ?

I.e. what is the probability of $X$ happening if $A$, $B$ and $C$ have happened given the probability of $X$ happening independently and the probabilities of $X$ happening if $A$, $B$ or $C$ have occured.

Answer 1

You cannot calculate $P(X \mid A,B,C)$ without more information

Suppose, for example, all of $P(X)$, $P(X \mid A)$, $P(X \mid B)$ and $P(X \mid C)$ are $\frac12$

• Possibly $X$ is completely independent of $A,B,C$ in any combination, so $P(X \mid A,B,C)=\frac12$

• Or possibly the following joint events each have probability $\frac14$ and are the only possible combinations, making $P(X \mid A,B,C)=1$:

  • $X,A,B,C$
  • $X,A^c,B^c,C^c$
  • $X^c,A,B,C^c$
  • $X^c,A^c,B^c,C$

• and there are many other possibilities