Expected crossings among ten random chords

Each of $10$ chords on a circle is drawn by picking both of its endpoints independently and uniformly on the circumference. On average, how many pairs of chords cross each other?

$15$. Two chords have four endpoints, and all three ways of matching those four points into two chords are equally likely; exactly one matching crosses, so a given pair intersects with probability $\tfrac13$. By linearity over the $\binom{10}{2}=45$ pairs, the expectation is $45/3=15$.