Consider the orthogonalized drunk who starts out at the...

Consider the orthogonalized drunk who starts out at the proverbial lamp-post: Each step he takes is either due north, due south, due east or due west, but which of the four directions he steps in is chosen purely randomly at each step. Each step is of fixed length $L$. What is the probability that he will be within a circle of radius $2 L$ of the lamp-post after 3 steps?

Key Concepts

Geometric Probability

Geometric probability links the count of favorable outcomes to the probability of the random walker being within a particular geometric region—in this case, a circle of a specified radius. It involves using geometric measures (like distance in Euclidean space) to assess probabilities, connecting the probabilistic behavior of the random walk with spatial constraints.

Euclidean Distance

The concept of Euclidean distance is used to calculate the straight-line distance between two points in the plane. For the walker’s position after several steps, one computes the net displacement by summing the individual vector steps and then applying the Euclidean norm. This calculation determines whether the walker’s final position lies within a given radius from the starting point, an idea that is critical in solving the problem.

Random Walk on a Lattice

This concept refers to a process in which a particle (or, in this case, the drunk) makes moves in discrete steps along a grid or lattice. At each step, the decision of which adjacent site to move to is determined randomly, usually with equal probability for each direction. This framework is central to the problem, as it defines the space of possible positions the walker can occupy after a series of steps.

Combinatorial Analysis

Combinatorial analysis involves counting the number of distinct sequences or pathways that lead to different outcomes. In the context of the random walk, it is used to enumerate both the total number of possible paths and the number of paths that result in the walker being within a certain distance from the starting point. This counting is essential for determining the probability of a specific event occurring.

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