A Brownian Motion Problem

Problem statement:

Figure 1.

A point is performing Brownian motion (a "drunkard's walk") in the plane. It is now at (0,1). It will eventually reach the x axis. When it next does, what is the probability that it will be at a point where x > 1?

Solution show

How to generalise this solution show

This is one of several pages on using symmetry in mathematics.