Catalan Numbers 1: Chord Intersections

Suppose $$2n$$ points are chosen uniformly at random along the perimeter of a circle and paired into $$n$$ chords. What is the probability that none of the chords intersect?

Answer

The total number of ways to pair $$2n$$ points is

(2n - 1)!! = 1 \cdot 3 \cdot 5 \cdots (2n - 1).

The number of non‐crossing pairings is the $$n$$ th Catalan number

C_n = \frac{1}{n+1}\binom{2n}{n}.

Hence the probability of no intersections is

\frac{C_n}{(2n - 1)!!}.

Notes and comments

Comment 1: See that for $$n=2$$ , $$C_2 = 2$$ and $(2\cdot2-1)!! = 3!! = 3$ , giving $$2/3$$ ; for $$n=3$$ , $$C_3 = 5$$ and $$5!! = 15$$ , giving $$5/15 = 1/3$$ , in agreement with the earlier results.