I suppose that
$$P(A|B) = P(A | B,C) * P(C) + P(A|B,\neg C) * P(\neg C)$$
is correct, whereas
$$P(A|B) = P(A | B,C) + P(A|B,\neg C) $$
is incorrect.
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?
Best Answer
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:
$P(A | B) = P(A | B, C) + P(A | B, \neg C) = 1 + 1 = 2$
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.