One day, Diaconis was approached by a man who wanted to physically roll a die one million times and compare the real-world results to the standard breakdown of die roll probabilities. Well, the man actually rolled the die three million times under Diaconis's watch...

As it turns out, after about ten thousand the die starts to get round. The roundness is due to the die hitting the rolling surface and having little bits chipped off, and at ten thousand this chipping becomes statistically significant. And so the mathematical models we have of many many rolls eventually regressing toward perfect distributions are basically impotent, because, duh, you can't roll a round die and expect to get good results.