Probability of the event knowing that I received no informations
The right denominator should be $P(\bar C)$, and the right numerator should be $P(A\cap \bar C)$ since $\bar C$ is the event of no receiving information, either because the guy is not there or the guy is there but gives no information. The denominator should not depend on the true state of his "coolness". 

An easy way to picture this problem is by doing a diagram. Think of a square and divide it in half, say vertically where the mass (or probability) in the left side represents $P(A)$ and complementary probability in the right. Then divide the square in another half, say horizontally, so that the mass in the top half represents $P(B)$. each of the resulting smaller squares contains the mass of each of the possible intersections between these two events. Finally, you can divide the top half (created in the second step) in half (say horizontally again) where that top part represents $P(C)$, and the bottom part (from this last line up to the bottom of the square) represents $P(\bar C)$. Clearly, $C$ does not intersect with $\bar B$ and $\bar C$ contains $\bar B$, as desired. And you can clearly visualize the probability you are trying to calculate.