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Each face of a cube is painted randomly one of the colors red, orange, yellow, green, blue, purple, brown, black, or white. What is the probability that the cube has at least one pair of faces that share an edge and are the same color? Express your answer as a decimal to the nearest thousandth.

 Jun 28, 2023
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Each face of a cube is painted randomly one of the colors red, orange, yellow, green, blue, purple, brown, black, or white. What is the probability that the cube has at least one pair of faces that share an edge and are the same color? Express your answer as a decimal to the nearest thousandth.    

 

The first face that we're comparing can be any of them,   

so all that needs to concern us is the second face.    

 

There are 4 faces that share an edge with the first face.  

All of them except the face directly opposite.  

 

There are 9 colors available, so the second face has a 1/9   

chance of being the same color as the first face.   

 

But there are 4 adjacent faces, so you have 4 shots at   

getting the same color.  This makes the probability 4/9.   

 

Expressed as a decimal fraction, 4/9 = 0.444.  

.

 Jul 7, 2023

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