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 Bird on a Wire (Posted on 2004-06-07)
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!

 No Solution Yet Submitted by Sam Rating: 3.7000 (10 votes)

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 re: Computer simulation | Comment 8 of 42 |
(In reply to Computer simulation by Charlie)

This isn't entirely correct. It seems to imply that the amount of painted wire increases as the number of birds increases from 2 to infinity. However, the expected value of the painted portion of 2 birds is (10/3) meters, and the expected value of 3 birds is (10/3)+(10/6) meters (5 meters) because both segments must be counter. However, from here it decreases, as we are not guaranteed that each segment will be counted (though it is possible). Thus, at infinity the expected amount of paint is (70/18), as shown by your simulation.
 Posted by Eric on 2004-06-07 14:55:47

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