Set theoretic notation for statistical sampling without replacement?

Question

I would like to know how to use set theoretic notation to define a subset $D$ of a finite event space $S$ that contains exactly $n$ elements $d_i$ which are sampled uniformly without replacement from $S$. I am struggling to develop the appropriate notation for this, could somebody help?

Answer 1

Fo sampling without replacement, the answer is just $$\{K\subseteq S| \text{card}(K) = n\}$$ This specifies the set of events. You also have to specify the probabilities of the events. I would simply says that all events have the same probability, or that they are equally likely.

Following up on the questions you asked it the comments, it's more complicated to describe the events for sampling with replacement. You use the cartesian product of the set with itself $n$ times, if the order in which the elements are drawn is significant, but if order doesn't matter you'd have to use multisets.

In practice, you would just say sampling with or without replacement, since anybody at all familiar with probability knows what these mean.