Three random chords on a circle avoiding crossings

Six points are placed independently and uniformly on the circumference of a circle. They are then grouped uniformly at random into three pairs, and each pair is joined by a chord. Find the probability that no two of the chords cross.

$1/3$. Only the cyclic order of the points matters. There are $5\cdot3\cdot1=15$ equally likely ways to pair six points, and the number of non-crossing pairings of six points in convex position is the Catalan number $C_3=5$, giving $\tfrac{5}{15}=\tfrac13$.