I suppose that
is correct, whereas
However, I have got an “intuition” about the later one, that is, you consider the probability P(A | B) by splitting two cases (C or Not C). Why this intuition is wrong?
Suppose, as an easy counter example, that the probability P(A) of A is 1, regardless of the value of C. Then, if we take the incorrect equation, we get:
That obviously can’t be correct, a probably cannot be greater than 1. This helps to build the intuition that you should assign a weight to each of the two cases proportional to how likely that case is
, which results in the first (correct) equation..
That brings you closer to your first equation, but the weights are not completely right. See A. Rex’ comment for the correct weights.