Sample Size Determination for Hypergeometric Rare Event Suppose that there are 10 smokers and 590 non-smokers in a room and let $X$ be a random variable that describes the number of smokers picked out, without replacement, in $n$ random draws. What methods could be considered for determining the minimum sample size for estimating that the population contains about 10 out of 590 smokers with some level of confidence? Are there any specific methods that perform better when the ratio 10/590 is small, say, about $10^{-2}$?

The most challenging part here is that the sampling is done without replacement, which complicates everything, if we want to stay precise.
If you know for sure that the number of smokers in the room of 600 people is 10 (a null hypothesis, which is known to be true, in a sense), you may want to first start thinking towards minimizing the probability of getting samples with zero smokers. The latter is in a sense a p-value for the lower bound of the two-sided confidence interval, if this bound is set in between 0 and 1.
This probability equals:
$$ \frac{590 \cdot 589 \cdot 588 \cdot ... \cdot (590-n)}{600 \cdot 599 \cdot 598 \cdot ... \cdot (600-n)} = \frac{590! \cdot (600-n)!}{600! \cdot (590-n)!}$$
You can see that, for instance, the p-value for this lower bound falls below 0.025 (if we target a 95% two-sided confidence interval) at $n=184$ (link to wolframalpha)
