Non-Intersecting Chords

Two points are chosen uniformly at random on the unit circle and joined to make a chord \(C_1\). This process is repeated \(3\) more times to get chords \(C_2, C_3, C_4\). The probability that no pair of chords intersect can be expressed as \(\frac{a}{b}\) where \(a\) and \(b\) are coprime positive integers. What is the value of \(a + b\)?

Details and assumptions

A chord of a circle is the line segment between 2 points on the circumference. It does not extend infinitely in either direction.

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