Seems too simple, so I'm probably misreading it. There are six 4x4 faces (yielding 16 1x1 faces each, after cutting) painted red. After cutting there are 64 cubes with six faces each, or 384. 96 of these are red (16 from each of the six original 4x4 faces), so 96 in 384, or 0.25.
If it seems too easy, I'm probably missing the boat somewhere. Where? Is choosing a cube and tossing it different from just selecting a face at random?