Probability that every polygon contains almost exactly the expected number of samples

Question

Let's uniformly sample $n$ times from the unit square. Define a polygon contained within the unit square with area $A$. Surely as $n \to \infty$, the probability that the polygon contains at least $\lfloor{nA}\rfloor$ samples and at most $\lceil{nA}\rceil$ samples tends to 1. What I'm interested in is if there are $m$ polygons, $m = kn$, where the polygons may overlap arbitrarily, does the probability that every polygon has at least $\lfloor{nA_i}\rfloor$ and at most $\lceil{nA_i}\rceil$ samples tend to 1 as well?

Answer 1

Your statement

Surely as $n \to \infty$, the probability that the polygon contains at least $\max(0, \lfloor{nA}\rfloor)$ samples and at most $\lceil{nA}\rceil$ samples tends to $1$

is incorrect. It would be correct to say that the number contained in the polygon divided by $n$ converges towards $A$ in a law-of-large-numbers sense, but that is not saying the same thing. Flip a fair coin $1$ million times, and the proportion of heads will probably be close to $\frac12$ but the probability you see exactly $500\,000$ heads is about $0.0008$.

The number sampled in the polygon has a binomial distribution with parameters $n$ and $A$, so expectation $nA$ and variance $nA(1-A)$ which is increasing with $n$. The probability that the numbered sampled in the polygon is somewhere from $\lfloor{nA}\rfloor$ through to $\lceil{nA}\rceil$ converges to $0$ as $n$ increases.

With $m$ polygons, the probability is of course lower that they are all in this interval than that one specific one is.