Can you please define random cuts? you are stepping into the problem in Bertrand's Paradox.
I had random endpoints in mind, but I suppose either of the other methods might make interesting problems too.
Disclaimer: since this is an original problem, I am not quite sure of my own solution.
I just tried solving the problem with the "random midpoint" and "random radius" methods. I think they may be easier (if I'm not wrong).
I encourage all of you to try solving the problem using all three methods outlined in Anza Power's link.
Whoever finds all three solutions wins a free compliment.