What is the probability that a randomly drawn chord will be longer than the radius of the circle?
Bryan and np_rt both give the correct solution for their respective ways of choosing a random chord.
However, I'd have to go with np_rt's solution as that of the more "natural" way of randomly drawing a chord.
Imagine a flat universe on a 2-D plane. Then imagine it being criss-crossed with either micrometeoroids or by cosmic rays or whatever at random locations. A circle could be intersected by any of these rays. Let the first one be the random chord that we're talking about. It's coming from some direction or other, and will cross a perpendicular diameter (and radius) of the circle at some random point uniformly distributed along that radius, the way np_rt posits. And, his derivation being correct, makes his solution the "more correct" one, IMHO. You can't get more natural, or more random, than where a cosmic ray will hit.
Bryan's depends on a person choosing two points on a circle and then making a chord to connect them. There's no natural analog to this method of choice of points, so I'd call it less random.
Posted by Charlie
on 2003-10-09 15:32:00