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!

(In reply to Concrete Answer is (n-1)/(n+1) by Richard)

The birds of course, being zero size themselves, do not "occupy the whole wire", but I assume what is meant is that the number of painted strips gets larger and larger.  However, the number of unpainted strips gets larger and larger also.  And since the portion painted is to the nearest neighboring bird, the unpainted strips are larger than the painted ones.

Posted by Charlie on 2004-06-08 20:26:58