# 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

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

- Geometric Probabilities
- Are Most Triangles Obtuse?
- Barycentric Coordinates and Geometric Probability
- Bertrand's Paradox
- Birds On a Wire (Problem and Interactive Simulation)
- Buffon's Noodle Simulation
- Averaging Raindrops - an exercise in geometric probability
- Rectangle on a Chessboard: an Introduction
- Marking And Breaking Sticks
- Random Points on a Segment

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