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© 1999-2010 by Francis Edward Su
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From the Fun Fact files, here is a Fun Fact at the Medium level:

Pass the Beer

Figure 1
Figure 1

Suppose you and a bunch of friends are sitting around a table. There are N of you. You have a jug of beer in front of you, which no one has yet tasted.

So you take a swig of it, and then pass it to your left or right with probability 1/2. Now suppose your neighbor does the same---she takes a swig of it and passes it to her left or right with probablity 1/2. Each player continues in this fashion.

Because the beer is moving back and forth randomly around the table, it may be a while before some people get to taste the beer for the first time.

Which person around the table is most likely to be the last one to try the beer? Is it a person near you or far from you? (Assume that the jug is bottomless, and never runs out.)

The surprising answer is that ALL participants (except the first) are EQUALLY LIKELY (probability 1/(N-1)) to be last!

Presentation Suggestions:
Poll the class before you tell them the answer. Draw a diagram on the board, mark the starting person, and then point to various other persons on the diagram and ask "How many think it is this person? Or this one?"

You'll find that most people think the answer is the person farthest away from the starting person.

The Math Behind the Fact:
Try calculating the probability for some specific cases: n=3 is trivial (1/2 each of the other 2 players). The case n=4 is a little more challenging. The general case can be proved by considering any fixed player and conditioning on the time when the beer first reaches one of his neighbors.

How to Cite this Page:
Su, Francis E., et al. "Pass the Beer." Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

Keywords:    random walk
Subjects:    probability
Level:    Medium
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
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