Drunken Walker and Fly

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>.

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