Painted small cube showing red twice

A $3\times3\times3$ cube is painted red on all its outside faces and then sliced into $27$ unit cubes. You pick one unit cube at random and roll it twice like a die; both times the face on top is red. What is the probability you picked one of the corner cubes?

$\tfrac47$. By Bayes, weight each type by its count times the chance of two red rolls: corners $8\cdot(\tfrac36)^2=2$, edges $12\cdot(\tfrac26)^2=\tfrac43$, face centres $6\cdot(\tfrac16)^2=\tfrac16$, and the centre cube $0$. The posterior is $2\big/\big(2+\tfrac43+\tfrac16\big)=\tfrac47$.