So suppose I have a $n \times n$ square, $R$, on the $x-y$ axis. I generate $k$ random points inside $R$ that are uniformly distributed throughout the square. Suppose also that I have another fixed point $(a,b)$ inside $R$.
My question is, how large should $k$ be to ensure that the nearest point (measured by Euclidean distance) from these $k$ generated points is at most $\epsilon$ away from $(a,b)$?
The probability that a single generated point lie inside the circle around $(a,b)$ of radius $\epsilon$ is, (assuming that the complete circle lies within $n \times n$ square),
$$p_{success} = p = \frac{\pi \epsilon^2}{n^2}$$
Define $X$ as a random variable following geometric distribution with success probability $p$. Here, $X$ represents the number of points that should be generated in order to get a success,
$$P(X=x) = (1-p)^{x-1}p , \ \ x \in \{1, 2, \ldots \}$$
So, the expected number of points will be $E(X) = \frac{1}{p} = \frac{n^2}{\pi \epsilon^2}$