We color every face of a cube red, white, or green. We choose the color of each face independently at random. Each color is chosen with equal probability. What is the probability that there will be opposite faces of the same color?
It is easier to determine the probability of the event that there will not be opposite faces of the same color. Let the faces of the cube in three pairs of opposite faces be denoted by $A$ and $A^{\prime}, B$ and $B^{\prime}$, and $C$ and $C^{\prime}$. It is obvious that under the conditions of the problem, the three events that $A$ and $A^{\prime}, B$ and $B^{\prime}$, and $C$ and $C^{\prime}$ are of different colors are independent and equally probable.

Therefore, if $p$ denotes the probability that two opposite faces are of different colors, then $p^{3}$ is the probability that all three pairs of opposite faces are of different colors. Thus, $1-p^{3}$ is the probability that there is a pair of opposite faces of the same color. We still need to calculate the value of $p$.

Fix the color of one face. The opposite face can only receive two colors, each with a probability of $1 / 3$. Therefore, whatever the color of one face, the opposite face will be of a different color with a probability of $2 / 3$. Thus, $p=\frac{2}{3}$ is the probability that two selected opposite faces will be of different colors.

Therefore, $1-p^{3}=1-\left(\frac{2}{3}\right)^{3}=\frac{19}{27}$ is the probability that there will be a pair of opposite faces of the same color on the cube.