Birds on a Wire

Subject: a probability puzzle
Date: Fri, 05 Oct 2001 19:37:48 -0700
From: Mark Galecki

Alex,

I like (some parts of) your cut-the-knot site. My favourite mathematics professor in college - Marcin Kuczma, Warsaw University, gave us once this problem. It is an advanced problem to prove, but it can be easily simulated on a computer, and then answer "guessed". Either way, the answer is simply amazing.

Take a wire stretched between two posts, and have a large number of birds land on it at random. Take a bucket of yellow paint, and for each bird, paint the interval from it to its closest neighbour. The question is: what proportion of the wire will be painted. More strictly: as the number of birds goes to infinity, what is the limit of the expected value of the proportion of painted wire, assuming a uniform probability distribution of birds on the wire.

Post it and let your readers puzzle over it - it requires advanced math to prove it, so maybe not many will be able to prove the answer, but a lot of people can write a simple program on a computer and simulate and try to guess the answer. If you post it, please include the name of my professor as the author.

If you want, I can tell you what the answer is, either now or after some period of time.

Mark Galecki


If you are reading this, your browser is not set to run Java applets. Try IE11 or Safari and declare the site https:///www.cut-the-knot.org as trusted in the Java setup.

birds on a wire


What if applet does not run?

(The applet runs a specified number of trials for every number of birds between the specified minimum and maximum values.)

There were four write-ups at the CTKExchange.

Geometric Probability

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