If I take a random sample of a population, is the sample within a specific subset of the population still random? Say I have a population, and take a random sample of that population. Later, I decide I'm only interested in a specific subset of the population. A subset of the random sample is found in the subset of the population, as you might expect. Is that sample subset still random, with respect to the population subset?
To give an example, say I want to estimate how many people in a town of 12,000 people have blue eyes. I take a simple random sample X of 1200 people. Later, I decide I'm only interested in blue-eyed estimates for people with brown hair. 10,000 of the people in the town have brown hair. 1000 of the people in sample X are part of those 10,000. 50% of the people in that 1000 have blue eyes. Can I now comfortably say that 50% of the 10,000 have blue eyes (disregard confidence interval issues, etc.)? Or did I introduce some bias that would invalidate the sample?
 A: Let the population have $N$ members and the subpopulation have $M$ members.  Suppose your simple random sample (SRS) includes $k$ subjects.  Assuming the sampling is without replacement, what characterizes an SRS of size $k$ is that every $k$-element subset of the population has an equal chance of being the sample.
Now, any $k$-element subset of the population can be partitioned into the $i$ elements from the subpopulation that appear within that subset and the remaining $k-i$ elements not from the subpopulation.  So, given any $i\le k$ and given any $i$-element subset $S$ of the subpopulation, there are
$$\binom{N-M}{k-i}$$
possible ways to augment $S$ with other elements to make it of size $k.$  All these different ways give rise to distinct $k$-element samples of the whole population and these correspond to all the ways in which $S$ can result from your two-step process of sampling the population and then taking the subsample of elements from the subpopulation.  Consequently, all $i$-element subsets of the subpopulation have equal chances of being your subsample.  As such, you have an $i$-element SRS of the subpopulation.
The only difference between this scenario and one in which you first identify the subpopulation and take a sample of size $i$ from it is that here the quantity $i$ is random, whereas when you specify $i$ beforehand, of course it is not random.  Sometimes this distinction matters, but in most cases it does not, because usually estimates, predictions, etc. are conditioned on the sample size.
