# Help required with using the correct model for sampling

I regularly encounter the following scenario: I am presented with a set, $\mathcal{S}$, of $i$ items.

I am assured that $d$ of the $i$ items possess a particular property, $d'$ - and that the rest, $(i-d)$, do not. So the items are in 2 piles: $d'$ and non-$d'$.

(For context, let's say that $i$ is known to be 1,000,000 and $d$ is said to be 30,000. )

I stipulate that the probability of any one member of $i$ also being a member of $d$ is independent of other members of $i$.

Consider the case where I wish to test this assurance by taking a sample. (For practical and financial reasons, I cannot test all of $\mathcal{S}$. )

Clearly I don't know how common the property of $d'$ actually is within $\mathcal{S}$.

1. What size of sample from $\mathcal{S}$ do I need, for a given confidence and margin of error to test whether the provided value of $d$ is correct or not? E.g., if I was told $d$ was 30,000 and my sample indicated it was 30,100 I would be happy. If the indication was that the real value was 40,000 I would not be happy.

2. Suppose I only have access to the items marked $d'$ and cannot sample the rest. I believe then I can only test for non-$d'$ items in the pile marked $d'$, what I call false positives. Can I then infer anything about false negatives, e.g. $d'$ items in the non-$d'$ pile which should not be there?

3. Suppose further that in fact some proportion of the items in $\mathcal{S}$ do have a dependency, such that if a particular item $p$ has $d'$ then item $p+1$ has $d'$ also . Could I reasonably remove the dependency by considering i and i+1 as a single item?

If someone could point me at the relevant area of statistics, I would be very grateful!