{"id":365,"date":"2019-06-26T23:46:29","date_gmt":"2019-06-26T23:46:29","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=365"},"modified":"2019-12-09T22:42:42","modified_gmt":"2019-12-09T22:42:42","slug":"pass-the-beer","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/pass-the-beer\/","title":{"rendered":"Pass the Beer"},"content":{"rendered":"\n
\"\"<\/figure><\/div>\n\n\n\n

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.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

Which person around the table is most likely<\/em> to be the last<\/em> 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.)<\/p>\n\n\n\n

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

Presentation Suggestions:<\/strong>
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?”<\/p>\n\n\n\n

You’ll find that most people think the answer is the person farthest away from the starting person.<\/p>\n\n\n\n

The Math Behind the Fact:<\/strong>
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.<\/p>\n\n\n\n

How to Cite this Page:<\/strong> 
Su, Francis E., et al. “Pass the Beer.” Math Fun Facts<\/em>. <http:\/\/www.math.hmc.edu\/funfacts>.<\/p>\n\n\n\n

Fun Fact suggested by: <\/strong>
Francis Su<\/p>\n","protected":false},"excerpt":{"rendered":"

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