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}

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About mvtrinh

Retired high school math teacher.
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