Jar A has exactly four red buttons and eight blue buttons.  Carla then removes the same number of red buttons as blue buttons from Jar A and places them in an empty Jar B.  Jar A now has $\frac{2}{3}$ of its original number of buttons.  If Carla were now to randomly choose a button from Jar A and a button from Jar B, what is the probability that the two chosen buttons will both be red?  Express your answer as a common fraction.
Two-thirds of Jar A's original $4+8=12$ buttons is 8 buttons.  Therefore, four buttons were removed from Jar A: two red buttons and two blue buttons.  So the probability that the button drawn from Jar A is red is $\frac{2}{8}$ and the probability that the button drawn from Jar B is red is $\frac{2}{4}$.  Therefore, the probability that both buttons are red is $\dfrac{2}{8}\cdot\dfrac{2}{4}=\boxed{\frac{1}{8}}$.