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Strike a Chord (..Any Chord) (Posted on 2003-10-09) Difficulty: 4 of 5
What is the probability that a randomly drawn chord will be longer than the radius of the circle?

Prove it.

No Solution Yet Submitted by DJ    
Rating: 4.5263 (19 votes)

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i think | Comment 14 of 51 |
i think the problem's a good one, but rather than finding the actual probability, the difficulty is in finding a truly random way of picking a chord. i don't think that selecting a single point on the circumference and then picking any other point is as good a method, because if you draw the chords on a circle, and draw the lines to evenly spaced points on the circumference, the chords you draw that way are ~scrunched~ at the bottom and ~spread out~ at the top. On the other hand, if you draw lines from a single point at evenly spaced angles, they are distributed differently. what you guys were saying about picking a point and then another point is actually equivalent to picking an angle, not a point, because you were considering the points to be evenly distributed by the angle they form, rather than arclength.
i'm not sure that makes sense, but i'm pretty sure they are two different analyses.
also, nobody has commented on the thing dj added, about picking a random point, which would work because each point in the circle maps 1:1 directly to a single chord -- every point except for the center. there are an infinit number of diameters, and of course the midpoint of each is the center of the circle. however, since the center makes up for only as much of the ratio as any other point, that would seem to imply that the actual probability is even greater than 3/4. then again, the center is part of both the inside and outside circles, so if the center actually should count greater than the other points, perhaps the probability is less (if a<b and c>0, (a+c)/(b+c)>a/b ).
i find this whole discussion very fascinating, although i don't think i'm explaining my thoughts very well. i look forward to seeing what everyone else has to say.
  Posted by Aaron on 2003-10-10 02:03:43
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