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 sameshe 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/(N1))
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>.
