A world record in Atlantic City and the length of the shooter's hand at craps

TL;DR

This paper analyzes the probability distribution of the length of a shooter's hand in craps, providing a closed-form expression and revealing it as a combination of four geometric distributions, with context from a notable world record.

Contribution

It derives an explicit closed-form expression for the distribution of the shooter's hand length in craps, advancing beyond recursive or Markov chain methods.

Findings

Distribution of shooter's hand length is a linear combination of four geometric distributions.

Provides an explicit formula for probability of long shooter's hands.

Analyzes a real-world record event to illustrate the model.

Abstract

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.