Assume a universe $U$ with $n$ elements. Let $p$ be the fraction of elements in $U$ with some property $P$. The goal of this problem is to estimate $p$ using samples from $U$. Given input parameters $\varepsilon$ and $\delta$, we want our estimate $\hat{p}$ to have the following guarantee: $\Pr[ |\hat{p} - p| \geq \varepsilon ] \leq \delta$.
Standard Approach: sample a set $S$ of size $O(\frac{1}{\varepsilon^2} \ln (\frac{1}{\delta}))$ uniformly at random and independently from $U$, and then set $\hat{p}$ to be the fraction of the sampled elements with property $P$. Using Chernoff bound, we can show that $\hat{p}$ has the desired guarantee.
Now consider the following sampling process. Sample each element in $U$ with probability $q = \frac{1}{\varepsilon^2} \ln (\frac{1}{\delta})\frac{1}{n}$. Let the resulting set be $S$. Note that $\varepsilon$ and $\delta$ are input parameters to the process. So if it happens that $q \geq 1$, then the sampling process would store the entire universe, i.e., $S=U$. So assume $n$ is large enough, such that $q < 1$. Now set $\hat{p}$ to be the fraction of the elements in $S$ with property $P$. Does $\hat{p}$ still have the same guarantee as before? If so, then how does one go about proving it? Any pointers to the existing work would also be very helpful.