# The Drunkard’s Walk and Catalan Numbers

In this post, I’d like to present an amusing and off-the-beaten-path solution to the classical “Drunkard’s Walk” problem which at the same time derives the well-known generating function for the Catalan numbers.  This solution evolved from a suggestion by my former undergraduate student Stefan Froehlich during a discussion in my Math 4802 (Mathematical Problem Solving) class at Georgia Tech in Fall 2007.

In case you’re not familiar with it, here’s the problem (in a formulation I learned from this book by Frederick Mosteller):

A drunken man standing one step from the edge of a cliff takes a sequence of random steps either away from or toward the cliff. At any step, his probability of taking a step away from the cliff is $p$ (and therefore his probability of taking a step toward the cliff is $1-p$). What is the probability that the man will eventually fall off the cliff?