Sampling and Conditions Suppose we consider all the people in the planet. We are interested in randomly selecting 10 people who have heart disease. Is it better to repeatedly sample 10 people and then choose the one selection in which all 10 people have heart disease? Or is it better to only look at the people who have heart disease and sample 10 from that population? 
 A: So long as it is truly random, and your hypothetical list covers everyone on the planet, then it would be most efficient to only select from the group with heart disease as that is what you are interested in studying. 
There shouldn't be any difference between randomly sampling 10 until you have 10 with heart disease and randomly sampling 10 from those that you know have have heart disease. This is assuming your list of those with heart disease includes everyone with heart disease on the planet.
A: The question of equivalence is settled by considering every possible outcome of the sampling procedure and determining its chance of being the sample.  When two procedures select all outcomes with the same probabilities, they are statistically equivalent.

In the first case, we repeatedly sample subsets of $10$ people out of all $N$ on the planet.  We do so in a way that selects every one of the $\binom{N}{10}$ distinct subsets with equal probability.  We repeat until the one we select is a subset of the $M\le N$ people who have heart disease.  (Let's call such a subset "desirable" and all other subsets "undesirable.")  That makes the procedure somewhat complicated: it is a process that can take an arbitrarily large number of steps.
One way to compute the chance of a particular set of diseased patients being selected is to break down the process according to how many steps were taken: that set was either selected on the first attempt, or on the second attempt, or, ..., or on the $K^\text{th}$ attempt, or ... .  The chance of being selected on the $K^\text{th}$ attempt is the chance of being selected on that attempt given that on every one of the preceding $K-1$ attempts, one of the $\binom{N}{10}-\binom{M}{10}$ undesirable subsets was selected.  Assuming the separate attempts were independent, this chance is computed by multiplying the chances at each attempt, giving
$$\left(\frac{\binom{N}{10}-\binom{M}{10}}{\binom{N}{10}}\right)^{K-1}\frac{\binom{M}{10}}{\binom{N}{10}}$$
for the chance of being selected on the $K^\text{th}$ attempt.  Summing these values for $K=1, 2, \ldots$ gives the chance of being selected.  Fortuately, we do not actually have to calculate this chance: it suffices to observe that it does not depend on the particular set we have under consideration, but only on the numbers $N,$ $M,$ and $10.$  Thus, the first sampling procedure selects all desirable sets with equal probability.  Obviously that's what the second procedure does (by design), so the procedures are statistically equivalent, even though they have been conducted differently.

This analysis ought to look like overkill, because it is: it should be obvious that all desirable sets have the same chance of being selected using either procedure, because no individual is favored in the selection nor are the selections of any individuals interdependent (as would happen if, for instance, entire households of people were selected at a time).  The reason for presenting this analysis in such detail is to demonstrate, with a simple example, how one might go about evaluating any sampling procedure of a defined population: namely, you determine the chance of every possible sample.  
Although the concept is simple, the results can be very illuminating in complex situations, such as hierarchical sampling or spatial sampling schemes.  For instance, a common way to sample soils in a field is to pick a random origin and a random orientation, lay out a regular grid of points starting at that origin and oriented accordingly, and taking a sample at each grid point that falls within the field.  By emulating the analytical process exemplified here, you will be led to discover that many possible sets of samples have no chance of being selected at all.  For instance, they won't all be in the same half of the field.  This provides valuable insight into why the usual statistical procedures are not valid to apply to such samples (except, occasionally, as approximations).
