Suppose I have a circle with circumference $A$. Along the circumference of this circle, I randomly drop $N$ arcs with fixed length $a < A$. Now suppose I drop a single additional arc ($N+1$). What is the probability $P(N, a/A)$ that this arc does not overlap with any previously dropped arcs?
My intuition is that this has something to do with the Stirling numbers, because given something like $3$ arcs, the overlap scheme could be no overlapping, three possible ways of three overlapping, or all overlapping. However, I cannot figure out how to approach the problem of finding the probability of overlap. I found some relevant notes on meeting probabilities here and here, but I can't quite see how to apply this to my problem