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Random chord on circle

Problem: 

Given a circle of radius 1, pick a chord at random. What is the chance that the length of the chord is less than the radius?


Solution:

Depending on how you pick it at random, the probability can be different!


1. If you pick it by randomly choosing 2 points on the circle's circumference, then the chance would be 1/3.

2. Every chord has a midpoint which is also a perpendicular bisector. If we pick this midpoint randomly within the circle, its chord length will be less than radius as long as it is outside of a smaller circle with radius $\frac{\sqrt{3}}{2}$. So the chance is 1/4 based on the ratio of area.

3. From the centre, randomly choose a radial direction and randomly pick a point along this radius, then draw a radially perpendicular line through this point, we'll get a chord. Then the chance would be \( 1 - \frac{\sqrt{3}}{2} \).

Surprising, huh. In fact, you can compute the distribution of the chord length given the picking method. Is there a way to pick a chord "uniformly"?

Another interesting and related problem: how do you uniformly pick a point within a circle? Can you prove that this method of picking a point is not uniform: first uniformly pick $\theta$ and then uniformly pick $r$, we get $x = r \sin \theta$, $y = r \cos \theta$?


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