Color and Roll
Jane paints the outer faces of a \(5\times5\times5\) cube green and then divides the cube into 125 smaller \(1\times1\times1\) cubes. A single cube is then selected at random and rolled. Calculate the probability that the upward face of the selected cube is green.
Answer
Notice that all the faces of the smaller cubes are equally likely to show as the up face. There are \(6\times 5^3\) faces in total. There are \(6\times 5^2\) colored faces. Hence, the probability that a colored face shows as the up face is:
\[
\frac{6\times 5^2}{6\times 5^3}
= \frac{1}{5}.
\]