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
This can be quite delicate but lots are known, going back to a classic paper by Flatto and Konheim available here.
Let $n-1$ uniform points be independently selected on the circle, and denote the the $n$ interval lengths they (with probability one, being distinct) split the circle into by $X_1,\ldots,X_n.$
For example, the following quantities are calculated in this paper and they can directly answer your question and similar questions.