jmc

There are three different possibilities for our first decision, each corresponding to which container we choose. So, if we choose container I, with $\frac{1}{3}$ probability, we have a $\frac{4}{12}=\frac{1}{3}$ probability for a $\frac{1}{3}\cdot \frac{1}{3}=\frac{1}{9}$ probability of getting green from Container I. Similarly for container II the probability is $\frac{1}{3}\cdot \frac{4}{6}=\frac{2}{9}$, and the same for container III. So, the total probability is $\frac{1}{9}+\frac{2}{9}+\frac{2}{9}=\boxed{\frac{5}{9}}$.