It's a paradox because esteemed mathematicians, including authors of classic books, have vociferously argued that their particular arbitrary choice is the sole

It's a paradox because esteemed mathematicians, including authors of classic books, have vociferously argued that their particular arbitrary choice is the sole correct choice. In probability, one must make many unstated assumptions to get an answer to a problem. Usually one believes those assumptions are "canonical", like uniform independent distributions. Sometimes woldviews contradict each other.