Figure 1

Imagine a drunken
person wandering on the number line who starts at 0, and
then moves left or right (+/1) with probability 1/2. What is the
probability that the walker will eventually return to her
starting point
Answer: probability 1.
What about a random walk in the plane, moving on the integer
lattice points, with probability 1/4 in each of the coordinate
directions? What's the chance of return to the starting
point?
Answer: also probability 1.
OK, now what about a drunken fly, with 6 directions to
move, probability 1/6? Surprisingly, it is probable that
the fly will never return to its start. In fact it only
has probability around 1/3 of ever returning.
Presentation Suggestions:
Try to give a little insight by illustrating a random walk
on the line for several steps.
The Math Behind the Fact:
A probabilist would say that simple random walks on the line and
plane are recurrent, meaning that
with probability one
the walker would return to his starting point, and that
simple random walks in dimensions 3 and higher are
transient, meaning there is a positive probability
that he will never return! This is because there is so
much "space" in dimensions 3 and higher.
How to Cite this Page:
Su, Francis E., et al. "Drunken Walker and Fly."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.
