A telephone wire stretched tight between two poles placed ten meters apart is a favorite resting spot for a flock of crows.

Suppose one morning two crows land on the wire, each at a random spot (the probability is uniformly distributed). With a bucket of paint and a brush you mark the stretch of wire between them. A certain length of wire will have been painted.

On average, what length of wire would you expect to have painted? Assume that each bird is a single point along the line, and so has no width.

Suppose instead that a dozen crows landed on the wire, each at an independent, random location, and you painted the stretch of wire between each bird and its nearest neighbor. On average, what total length of wire would you expect to have painted now?

And if a thousand crows landed?

A computer-generated solution could be found, but bonus points will be awarded for a formal proof!

The answer is 5m.  Because the point of landing of the birds is random and there is no spatial dimension to the point of landing.  The answer requests an average distance.  SO, the birds can be infinately close or infinately seperated along the 10m wire, an average of which can be assumed that the average length painted will be 5- no matter the number of birds.

Posted by Rob on 2004-06-13 01:05:30