I would put it in this way: Merely saying "random" is insufficient information.
To solve the problem, you have to know from what distribution chords are drawn. It could be equal probability per unit angle, per unit distance along the circle, or something else altogether.
Similar difficulties emerge when someone gives you a problem that starts with a "random real number."
All real numbers have an infinite number of digits. For the numbers that you probably meant to exclude, all but finitely many of those infinitely many digits are 0.
(This is almost just nitpicking, but in another sense it's important: viewing an apparently terminating decimal expansion as actually 0-padded lets one say correctly that "a real number is rational if and only if its decimal expansion eventually repeats", full stop, rather than having to add the qualification "… or is eventually 0.")
Well, to be sure, one can say that it does count or that it doesn't—the axioms of mathematics are only as humans make them. I was arguing why it might make sense to have it count.
Another reason that it might be a good reason to have it count: by my definition, I can say "the decimal expansion of a number eventually repeats if and only if the binary expansion of that number eventually repeats." This becomes considerably more awkward if you don't consider a terminating expansion as consisting of eventually repeating 0's; for example, 1/5 has a terminating decimal expansion 0.2 but an immediately-repeating binary expansion 0.repeat(0011).
To solve the problem, you have to know from what distribution chords are drawn. It could be equal probability per unit angle, per unit distance along the circle, or something else altogether.
Similar difficulties emerge when someone gives you a problem that starts with a "random real number."