An absent-minded professor goes for a walk carrying a digital audio device using 2 batteries. He has 2 fresh replacement batteries stashed away in one of four pockets. Sure enough, both batteries lose their charge and he removes them. Not wanting to throw the depleted batteries into the woods, he places them into a pocket chosen at random from the 4 available. A little while later he remembers the two fresh batteries, but he cannot remember which pocket. He fishes around in his pockets until he finds one with batteries (either 2 or 4 indistinguishable batteries). He removes 2 batteries and inserts them into the digital audio device. The digital audio device requires at least one good battery in order to play. Find the probability the digital audio device works on the first try.
I believe the denominator should be $(4)_2$, but I am having trouble with what the numerator should be.