Based purely on intuition, it would seem that the following statement is true, when thinking of the events as sets:
$$P(A\cap B) + P(A\cap B^c) = P(A)$$
However, I am not sure if this is true, and cannot find out how to prove it, or describe a straight forward intuition of it.
Is this equation true? And if not, is there another way to link $P(A\cap B)$ and $P(A\cap B^c)$ with $P(A)$?