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

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

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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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