Hidden face of a painted unit cube

A $3\times3\times3$ cube built from $27$ white unit cubes has its entire outer surface painted red and is then taken apart. You pick one of the $27$ unit cubes at random and set it down in a uniformly random orientation, so that you can see five of its faces and one face is hidden. All five visible faces are white. What is the probability that the hidden face is red?

$1/2$. Only two kinds of cube can show five white faces: the single unpainted centre cube (always) and the $6$ face-centre cubes with one red face (when that face is hidden, probability $1/6$). These give $\tfrac{1}{27}$ and $\tfrac{6}{27}\cdot\tfrac16=\tfrac{1}{27}$, equally likely, so the answer is $\tfrac12$.