## Red or Blue Cube

A cube’s faces are colored either red or blue. Each occurrence has a probability of $0.50$. The color of each face is determined independently. If the painted cube is placed on a horizontal surface, what is the probability of $4$ vertical faces having the same color?
Source: mathcontest.olemiss.edu 12/3/2012

SOLUTION

We label the six faces of the cube as follows
$A$ opposite $B$
$C$ opposite $D$
$E$ opposite $F$
A possible net of the cube shows the top face $A$, the bottom face $B$, and the four vertical faces $C,F,D,E$

Let $R$ represent the red color and $B$ the blue color. When face $A$ is on top, there are $2\times 2\times 2\times 2=16$ possible choices of color for the four vertical faces $C,F,D,E$
$RRRR$
$RRRB$
$RRBR$
$RRBB$
$RBRR$
$RBRB$
$RBBR$
$RBBB$
$\;$
$BRRR$
$BRRB$
$BRBR$
$BRBB$
$BBRR$
$BBRB$
$BBBR$
$BBBB$
Of the $16$ choices only $RRRR$ and $BBBB$ have the same color.
When face $A$ is on top, the number of possible outcomes equals $16$ and the number of desirable outcomes equal $2$. In fact, the same is true for all $6$ faces of the cube
Number of total possible outcomes = $6\times 16=96$
Number of total desirable outcomes = $6\times 2=12$
Probability of $4$ vertical faces having the same color = $\frac{12}{96}=\frac{1}{8}$

Answer: $\frac{1}{8}$