I don't quite know where to begin with a solution, so I'll articulate the problem.
"A right cylinder of uniform density has a radius, r, and a height, or thickness, h. If r is sufficiently larger than h, as with a coin, and the object is tossed about, the probability of it coming to rest on a base are very high. If r is relatively small, as with a caber, it's likely to come to rest on its lateral face.
If r is 1, what is the value of h such that, if the object is tossed and allowed to come to rest on a face, the probability of it resting on its lateral face are 0.333?"
Now, I didn't say "land on its lateral face" because I understand that an object could land on one face and then fall / roll into another, if that second face drops the object's center of gravity, and if there's sufficient momentum to raise the center of gravity due to the rolling itself.
The only value for h that doesn't cause any rolling at all would be h = 2, where the side-view of the cylinder is resembles a square. But that seems too large a value. The probability of such an object landing on its lateral face seems much higher than 0.333.
As i say, I'm stumped.