A wooden cube - after being painted all over - has been cut into smaller cubes. These were thoroughly mixed in a bag, from which one was produced and tossed. What is the probability that a painted side turned up?
On impulse, one would approach the problem in a more or less standard way. There are corner cubes with painted sides, mid-edge cube with painted sides, and face-central cubes with only side painted. The total probability is then
Now generalize: cut the cube into , smaller cubes and ask the same question. The problem is not awfully difficult but needs some figuring out. Following the foregoing pattern, we eventually arrive at
This expression simplifies, as you can verify, to which, by the way, confirms the answer for
The above is a useful and not too difficult exercise, but there is a delightful shortcut that avoids most of the counting of cubes and their sides.
cubes have a total of sides. Of these, are painted. All sides have the same probability of turning up, therefore, a painted side will turn up with the probability