A cube is cut into 1000 pieces. Since the volume of the total cube would be same as that of the smaller cubes, you can imagine that in each of the dimensions of the cube, length, breadth and height, there will be 10 cubes that would be incorporated.
Probabaility that a cube is colored on 3 sides -- such cubes would always be the corner cubes of the bigger cubes. And since, a cube has 8 corners,
Prob(3 sides colored) = 8/1000 = 0.008
2. Probability that the cube is colored on 2 sides -- such cubes would be on all the edges of the bigger cube except for the corner pieces on both sides of the edge (which would be colored 3 sides).
So, there would be 10-2 = 8 cubes on each edge which would be colored on 2 sides
and there are 12 edges in a cube, so totoal number of 2 side colored cubes = 12*8 = 96.
Hence Prob (2 sides colored) = 96/1000 = 0.096
3. Probability that a cube is 1 side colored would mean all the cubes obtained from the face of the larger cube, except for those on the edges. So, on any face, you would have 10*10 = 100 cubes being cut, out of which 36 would be on the edges. So, left are 100-36 = 64 cubes.
Now, there are 6 faces of a cube, so, total cubes with 1 side colored = 6*64 = 384
Hence, prob(1 side colored) = 0.384
4. Not colored = 1000 - 1 side colored - 2 side colored - 3 side colored = 1000 - 384 - 96-8 = 512
Hence prob(not colored) = 512/1000 = 0.512