Inner cube or face-down painted cubelet

A plain white $3\times3\times3$ cube is dipped in red paint so its whole exterior is red, then split into its 27 unit cubes, which scatter randomly. You examine one that has come to rest on a face, and none of the five faces you can see is red. What is the probability it is the cube from the very middle?

$1/2$. The visible faces are all white only for the central cube (always) or for one of the 6 cubes with a single red face resting red-side down (probability $\tfrac16$ each). Bayes gives $\tfrac{1/27}{1/27+6/27\cdot1/6}=\tfrac12$.