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Red Rubik's cube

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Parts

Part 1

A standard white 3 x 3 x 3 rubik’s cube is dipped into paint so that its entire external surface is colored red. Then, we explode the rubik’s cube, sending all 27 unit cubes flying everywhere. We inspect one in particular, which is lying face down. No faces are red.

What is the probability that this cube was the center one?