I'm currently doing some mathematical analysis of a system and I have distilled the problem down to a geometry/probability problem. I have a two part problem I would like to solve.
Part 1: Random Points on a Disk
From here, the probability density function of two points (randomly picked, uniformly) on a disk of radius $R$, having a distance between them of at most $s$ is:
$ f(s)=\frac{4s}{\pi R^2}\arccos\frac s{2R}-\frac{2s^2}{\pi R^3}\sqrt{1-\left(\frac s{2R}\right)^2}$
I'd like to adapt this to solve it for $n$ points. The question is this: Given $n$ points randomly distributed across a disk of radius, R, what is the probability that at least two of those points are within $s$ of each other?
My Guess: Take the integral of the above function from $0$ to $s$, and then multiply it by $^nC_2$ . However, this seems too simple. Am I mistaken?
Edit: As saulspatz below stated, this can't be right, because as $n$ increases, $\binom{n}{2}$ goes to $\infty$ , so I'm not sure how to solve this either.
Part 2: Total Expected Area of Overlap
Similar as above, but assume that the $n$ points selected all correspond to centers of circles with radius $r$. What is the expected area of overlap in terns of $n$, $R$ and $r$?
I did a bit of searching and there were a few topics that touched on this, but not really in the scope I was looking for:
Expected area of the intersection of two circles
Expected overlap of n circles of equal area randomly placed inside a circle of larger area
From what I could gather, this is a two part problem.
Part 2a: Area of Intersection of Two Circles
From here I found that the area of intersection of two circles with radii $r_1$ and $r_2$ at a distance $d$ from each other is as follows:
$A_{\textrm{intersection}} = r_1^2 \arccos\left(\frac{d_1}{r_1}\right) - d_1\sqrt{r_1^2 - d_1^2} \nonumber + r_2^2\arccos\left(\frac{d_2}{r_2}\right) - d_2\sqrt{r_2^2 - d_2^2}$
where:
$d_1 = \displaystyle\frac{r_1^2 - r_2^2 + d^2}{2d}$
and
$d_2 = d - d_1 = \displaystyle\frac{r_2^2 - r_1^2 + d^2}{2d}$
Because I'm only interested in circles of the same diameter, $r_1 = r_2 = r$ which can lead to some simplification of the equation above. I'll skip out on these simplifications for now.
Part 2b: Calculating Expected Areas
I'm not sure how to go about doing this here. After I simplify the above, clearly, I'm interested in looking at something $0 < d < 2r$ (if the distance between the two centers were more than $2r$, they would not intersect).
My Guess: I'd have to take some kind of integral of a function which is the product of the probability function $f(s)$ multiplied by the intersect area (setting $d=s$)
But those are only guesses from me. I'm not really sure if any of my guesses are correct (or even in the right direction). Could someone help?
Thank you
You are unlikely to find a closed form expression for the probability of overlap and general $n$. Also, note that that for all $n$ sufficiently large there will be overlaps with probability $1$. Indeed, $n$ disjoint circles of radius $r$ will have a total area of $n\pi r^2$, which exceeds $\pi R^2$ whenever $n>(R/r)^2$.
It is likely that for your application, a sufficiently good approximation of the probabilities in question will be enough, in which case I refer you to the literature in the well-developed field of random spatial models where extremely similar problems are studied.