Simple random sample

A simple random sample of size $n$, usually shortened to SRS, comes from a method under which every possible sample of $n$ individuals is equally likely to be the one drawn. The condition is about whole samples, not about individuals. Equal chance for each individual follows from it but does not imply it, and that gap is where most naming mistakes live.

With 5 students and $n = 2$ there are 10 possible samples, each of probability $1/10$, and each student sits in 4 of the 10, so every individual has chance $4/10 = 2/5$. Now take a different plan for four students A, B, C and D: flip one fair coin, take A and B on heads, take C and D on tails. Every student still has chance $1/2$, and the sample size is still 2, but only 2 of the 6 possible pairs can ever come up. The pair A and C has probability 0. The plan uses chance, and it is not an SRS.

So distrust the sentence "everyone had an equal chance of being picked, so it was a simple random sample." Equal individual chance is necessary and not sufficient. A cluster sample gives every individual the same chance while making most subsets impossible, which is why the SRS condition is written in terms of samples.

Carrying one out needs a list. Number a roster of 500 students 1 to 500, generate random integers in that range, ignore repeats, and take the first 30 distinct labels. If the roster leaves people out, the result is an SRS of the sampling frame and still biased for the population, because no amount of randomness inside a short list reaches anyone who is not on it.

Random selection is topic 1.11 Random Sampling in Unit 1. It is what supports generalizing to the population; random assignment is the separate move that supports a cause-and-effect claim.