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 1 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." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

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