A World Record in Atlantic City and the Length of the Shooter’s Hand at Craps

I t was widely reported in the media that, on 23 May 2009, at the Borgata Hotel Casino & Spa in Atlantic City, Patricia DeMauro (spelled Demauro in some accounts), playing craps for only the second time, rolled the dice for four hours and 18 minutes, finally sevening out at the 154th roll. Initial estimates of the probability of a run at least this long (assuming fair dice and independent rolls) ranged from one chance in 3.5 billion [5] to one chance in 1.56 trillion [10]. Subsequent computations agreed on one chance in 5.6 (or 5.59) billion [2, 6, 9]. This established a new world record. The old record was held by the late Stanley Fujitake (118 rolls, 28 May 1989, California Hotel and Casino, Las Vegas) [1]. One might ask how reliable these numbers (118 and 154) are. In Mr. Fujitake’s case, casino personnel replayed the surveillance videotape to confirm the number of rolls and the duration of time (three hours and six minutes). We imagine that the same happened in Ms. DeMauro’s case. There is also a report that Mr. Fujitake’s record was broken earlier by a gentleman known only as The Captain (148 rolls, July 2005, Atlantic City) [8, Part 4]. However, this incident is not well documented (specifically, the exact date and casino name were not revealed) and was unknown to Borgata officials. In fact, a statistical argument has been offered [4, p. 480] suggesting that the story is apocryphal. Our aim in this article is not simply to derive a more accurate probability, but to show that this apparently prosaic problem involves some interesting mathematics, including Markov chains, matrix theory, generating functions, and Galois theory.