We first need to determine how the 27 unit cubes are painted.
Number of Painted Sides | Number of Cubes |
3 | The 8 corner cubes |
2 | 12 non-corner edge cubes |
1 | 6 visible center cubes |
0 | 1 non-visible center cube |
Cases:
Case 1: We select a cube with 0 painted sides
It is not possible to see a painted side, since none are painted.
Case 2: We select a cube with 1 painted sides
The roll is successful as long as an unpainted side is on the bottom.
Case 3: We select a cube with 2 painted sides
The roll is successful as long as a painted side is on the bottom.
Case 4: We select a cube with 3 painted sides
It is not possible to see only one painted side, since only on can be hidden, but the other two cannot. Therefore, we will see more than one painted side.
Probabilty:
We only need to consider cases 1 and 2.
Case 2:
6 sides to a cube, 5 unpainted and 1 painted.
We need an unpainted side on the bottom, 5/6.
Case 3:
We need a painted side of the bottom, 2/6.
\(\frac{12}{27}\cdot\frac26+\frac{6}{27}\cdot\frac56=\boxed{\frac13}\)
I hope this helped,
Gavin