A white face on a painted cube

Idea. Count unpainted faces. Every unpainted face of every small cube is equally likely to be the one on top, so the answer is the centre cube's share of all unpainted faces.

Step 1: the kinds of cubes.

• 8 corners, with 3 painted faces and 3 unpainted;
• 12 edge cubes, with 2 painted and 4 unpainted;
• 6 face centres, with 1 painted and 5 unpainted;
• 1 centre cube, with 6 unpainted.

Step 2: unpainted faces. In total $$\begin{array}{rl}& 8\cdot 3+12\cdot 4+6\cdot 5+1\cdot 6\\ & =24+48+30+6=108.\end{array}$$

Step 3. A random cube and a random face make each of the $27\times 6$ faces equally likely to be on top. Given that the top face is unpainted, it is one of the 108 unpainted faces, and 6 of them belong to the centre cube: $$\frac{6}{108}=\frac{1}{18}.$$

A variant. If you see five unpainted faces (all but the bottom), the cube is either a face centre or the centre cube, with weights $6\cdot 1$ and $1\cdot 6$; the hidden face is then painted with probability $\frac{1}{2}$.