It was widely reported in the media that, on 23 May 2009, at the Borgata
Hotel Casino & Spa in Atlantic City, Patricia DeMauro, playing craps for only
the second time, rolled the dice for four hours and 18 minutes, finally
sevening out at the 154th roll, a world record. Initial estimates of the
probability of this event were erroneous, but consensus was reached within
days: one chance in 5.6 billion. More generally, what is P(L \ge n), where the
random variable L denotes the length of the shooter's hand (154 in Ms.
DeMauro's case) and n is a positive integer? It is well known that these
probabilities can be derived recursively or by Markov chain methods. Our aim
here is to give an explicit closed-form expression for them, showing that the
distribution of L is a linear combination (not a convex combination) of four
geometric distributions.
