Random Assignment vs Simple Random Sample

Random assignment means a chance device, not the experimenter, decides which unit gets which treatment, and the device's probabilities do not depend on anything about the unit. That is the whole requirement. It does not demand equal group sizes and it does not promise that the finished groups will look alike. What it buys is that every variable other than the treatment was spread across the groups by that same chance mechanism, so a difference in the response has only two explanations left, the treatment or chance, and the p-value measures the second one.

Balance is a tendency, not a guarantee. Take 20 subjects, 10 of them women, split into two groups of 10 by shuffling names. The expected split is 5 and 5, but the chance of landing exactly there is $\frac{\binom{10}{5}^2}{\binom{20}{10}} = \frac{63504}{184756} = 0.344$. A group holding 8 or more of the 10 women turns up about 2.3 percent of the time, and all 10 landing together about once in 92,000 assignments.

So "the treatment group came out older on average, so the randomization failed" reads the wrong thing. Random assignment is judged on the procedure used, not on the split it produced, and imbalance of exactly that size already sits inside the reference distribution the p-value comes from. Redrawing until the groups look even destroys that: the assignment is no longer random and the stated error rate no longer holds.

The boundary is who is in the study at all, and random assignment says nothing about it. Forty volunteers randomly assigned support a causal claim about people like those volunteers and about no one else. Widening the audience takes random selection, a separate act on a separate list; see scope of inference.

Topic 1.13 lists random assignment beside comparison, replication, and direct control as the four elements of a well-designed experiment.

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.