Corner cube given a red face on a painted cube

A $3\times3\times3$ white cube has its entire outer surface painted red and is then cut into $27$ unit cubes. One unit cube is chosen at random and rolled like a die, and the face that lands up is red. What is the probability that the chosen cube was a corner piece?

$4/9$. Every painted face is equally likely to be the one showing, so condition on the $6\cdot9=54$ red unit faces. The $8$ corner cubes carry $3$ each, i.e. $24$ of them, giving $\tfrac{24}{54}=\tfrac49$.