A cube 4 units on each side is composed of 64 unit cubes. Two faces of the larger cube that share an edge are painted blue, and the cube is disassembled into 64 unit cubes. Two of the unit cubes are selected uniformly at random. What is the probability that one of two selected unit cubes will have exactly two painted faces while the other unit cube has no painted faces?
The probability that a cube has exactly two painted faces is 1/16. This is because if two adjacent sides are colored, then only the 4 cubes along the edge of those two sides are painted twice. 4/64 is 1/16.
The probability that a cube has no painted faces is 9/16. This is because if two adjacent sides are painted, then a total of 16 + 12= 28 cubes are painted on at least one side. We can also see this by subtracting 4 (the cubes overlapping) from 32 total cubes on two faces.
We multiply these two probabilities together because they are dependent events to get 9/256