Is a random chord longer than the radius?

Question

A chord is drawn at random in a circle of radius $r$ .

What is the probability that the chord is longer than the radius?

Careful: this question does not have a single answer until you pin down how the chord is drawn. Give the answer under the three standard interpretations and explain why they differ.

Accepted Answer

Take r = 1; we want P(	ext{length} 1). Method 1, random endpoints. Fix one endpoint on the circle and choose the other uniformly on the circumference. If 2\alpha is the central angle between them, then \alpha is uniform on [0,\pi) and the length is 2\sin\alpha. Require 2\sin\alpha 1 \iff \sin\alpha 	frac12 \iff \alpha \in \left(	frac{\pi}{6}, 	frac{5\pi}{6}ight). That interval has length 	frac{4\pi}{6} out of \pi, so P1 = \frac{4\pi/6}{\pi} = \frac{2}{3}. Method 2, random radial midpoint. Pick a radius at random, then pick the chord's midpoint uniformly along that radius, drawing the c…

Is a random chord longer than the radius?

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