I am reading a book titled QBism, a description of the Bayesian interpretation of Quantum mechanics, The author describes the “... baffling cube factory paradox...”. A factory makes perfect cubes with edges randomly distributed between 0 and 1 centimeters. If you are given a bin of these and measure the edge length, what is the probability of getting a length less than 0.5 cm. Most people say 1/2 for this probability as 0.5 cm is in the middle of the range (they assume a uniform distribution of edge length). The “paradox” arises because the question might have been what is the probability that a face produced by the factory would have an area less than 0.5*0.5 cm^2 = 0.25 cm^2. The claim is that it would be 1/4 as the measures value is “1/4 of the total range or areas”.
As I see it, this is not even close to being a paradox. The problem is that nobody specified the distribution of the random process governing the cube manufacture. So no wonder we don’t get a well defined answer. The book kind of says this in the space of a couple of pages but I still don’t understand why this story is anything but trivial. Perhaps the full message is that you need to know the actual distribution before answering questions.
Here is the question: can anyone give an example where has this well-known paradox ever been used to illuminate any non-trivial issue? My research into this used google and I was not able to find anything, just descriptions similar to the above text, but my experience with probability theory is not extensive.