A cube’s faces are colored either red or blue. Each occurrence has a probability of . The color of each face is determined independently. If the painted cube is placed on a horizontal surface, what is the probability of vertical faces having the same color?

Source: mathcontest.olemiss.edu 12/3/2012

**SOLUTION**

**
**We label the six faces of the cube as follows

opposite

opposite

opposite

A possible net of the cube shows the top face , the bottom face , and the four vertical faces

Let represent the red color and the blue color. When face is on top, there are possible choices of color for the four vertical faces

Of the choices only and have the same color.

When face is on top, the number of possible outcomes equals and the number of desirable outcomes equal . In fact, the same is true for all faces of the cube

Number of total possible outcomes =

Number of total desirable outcomes =

Probability of vertical faces having the same color =

**Answer**: