Chords chords chords

Discrete Mathematics Level 3

In the circumference of a circle of radius \(r\), two points are chosen with a uniform distribution. The probability that the length of the chord between these two points is less than \(r \) is \( \frac{ a}{b} \) for coprime positive integers \(a, b \).

Find \(a+3b\).

This problem was from CB Paul Science quiz (Stage round) where it is expected to be solved in 40 seconds.
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