What is the probability that a randomly drawn chord will be longer than the radius of the circle?

Prove it.

Given a chord AB (where A and B are the chord's intersections with the circle), draw the triangle ABC where C is the center of the circunference.

If the angle ACB is less than 60 degrees, or more than 300 degrees, AB will be less than the radius. If the angle ACB is equal to 60 degrees or 300 degrees, AB wil have exactly the radius' length. If the angle is between 60 and 300 degrees, then AB will be longer than the radius. This range covers 2/3 of the angles' possibilities, so the probability of a chord being longer than the radius is 2/3.