easy probability question

Hello! Probability is troubling me. excuse me for the stupid thing I might be asking..:p

So, let's say there are two containers, one with 3 red balls and 7 black balls; the otehr with 6 rad balls and 4 black balls. If I am asked to compute the probabiulity to pick a red ball, intuitively I would sum all the red balls and divide by the total number of balls. Using the formula, I should say P(R)= P(R /\ C1) + P(R/\C2) where C stands for container and /\ intersect... ok but why should I do in this way? I mean, P(R given C1) + P(R given C2) is not correct, but I see it is not correct with numbers, because otherwise I would interpret it as the sum of the probabilities that I pick a red ball from container 1 plus the prob I pick a red ball from container two. But the second formula evidently does not mean this...so what does it mean? Does it have any sense?

Thanks a lot for you help.