Is Marginalization Always Valid for Joint Probabilities?

[tmt1]

Given $$P(B \land C)$$ will it always be true that $$P(B \land C | A) P(A) + P(B \land C | \lnot A) P( \lnot A)$$ (regardless what $P(A)$ would be)?

How can I prove this?

 

[I like Serena]

Given $$P(B \land C)$$ will it always be true that $$P(B \land C | A) P(A) + P(B \land C | \lnot A) P( \lnot A)$$ (regardless what $P(A)$ would be)?

How can I prove this?

Hi tmt, (Smile)

We have from set theory:
$$P(B \land C) = P(B \land C \land (A \lor \lnot A))
= P((B \land C \land A) \lor (B \land C \land \lnot A))
$$
Since $A$ and $\lnot A$ are mutually exclusive, as are subsets of them, it follows from the sum rule that:
$$P((B \land C \land A) \lor (B \land C \land \lnot A)) = P(B \land C \land A) + P(B \land C \land \lnot A)
$$
Then, from the general product rule, it follows that:
$$ P(B \land C \land A) + P(B \land C \land \lnot A) = P(B \land C \mid A)P(A) + P(B \land C \mid \lnot A)P(\lnot A)
$$
So indeed, without knowing anything about $P(A)$, we can state that:
$$P(B \land C) = P(B \land C \mid A)P(A) + P(B \land C \mid \lnot A)P(\lnot A)$$