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{\Large \bf DIMACS EDUCATIONAL MODULE SERIES}

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{\Large \bf MODULE 04-1}\\

{\Large \bf Probability and Chip Firing Games}\\

{\large \bf Date prepared: \today}

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{\bf Lynne L. Doty}\\
{\bf Marist College}\\
{\bf Poughkeepsie, NY 12601}\\
{\bf email: Lynne.Doty@Marist.edu}\\[1cm]

{\bf K. Peter Krog}\\
{\bf Marist College}\\
{\bf Poughkeepsie, NY 12601}\\
{\bf email: Peter.Krog@Marist.edu}\\[1cm]

{\bf Tracey Baldwin McGrail}\\
{\bf Marist College}\\
{\bf Poughkeepsie, NY 12601}\\
{\bf email: Tracey.McGrail@Marist.edu}

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{\large \bf Module Description Information}

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\begin{itemize}

\item {\bf Title:}

Probability and Chip Firing Games

\item {\bf Authors:}

Lynne L. Doty, Marist College, Poughkeepsie, NY 12601\\
K. Peter Krog, Marist College, Poughkeepsie, NY 12601\\
Tracey Baldwin McGrail, Marist College, Poughkeepsie, NY 12601

\item {\bf Abstract:}

This module investigates the use of chip firing games on finite 
directed graphs as a ``probabilistic abacus'' that can be used to 
solve probability questions that might otherwise appear to require 
the use of complicated infinite geometric series.  The use of chip 
firing games in this context leads naturally to the introduction of 
Markov analysis.

\item {\bf Informal Description:}

Probability questions arising from experiments or games of chance 
can often be difficult to solve due to the complicated nature of the 
collection of possible outcomes.  In certain circumstances, there 
might be a way to organize what could be an infinite collection of 
outcomes into a finite form that would offer alternate means of 
analysis.  This module addresses these issues and offers several 
different approaches to solving probability questions.

\item {\bf Target Audience:}

The intended audience consists of college students enrolled in a 
course on discrete mathematics or probability at the sophomore or 
junior level.

\item {\bf Prerequisites:}

Students should be familiar with infinite geometric series and basic 
discrete probability, including expected value.  Section 5 of the 
module requires that students also be familiar with basic matrix 
operations.

\item {\bf Mathematical Field:}

Probability, Chip Firing Games, Markov analysis.

\item {\bf Applications Areas:}

Most of the applications in this module deal with games, including 
both games of chance and children's board games.  There is also an 
application to genetics in Section 5.

\item {\bf Mathematics Subject Classification:}

Primary Classification:  91A43\\
Secondary Classifications:  91A60, 60J05

\item {\bf Contact Information:}

K. Peter Krog\\
Department of Mathematics\\
Marist College\\
3399 North Road\\
Poughkeepsie, NY 12601\\
email:  Peter.Krog@Marist.edu

\item {\bf Other DIMACS modules related to this module:}

None at this time.

\end{itemize}

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