= Simple random sampling
{title2=$\mathbb P(A=a)=\binom Nn^{-1}$}
= Simple random sampling without replacement
{synonym}
= SRSWOR
{c}
{synonym}
Here <simple random sampling> means a random unordered <subset> $A$ of exactly $n$ distinct labels from $\{1,\ldots,N\}$, with every size-$n$ <subset> equally likely. The <sample size> is fixed in advance, $1\leq n\leq N$, and <sampling without replacement> prevents duplicate units. This is <SRSWOR>; some texts also use "simple random sampling" for independent uniform draws with replacement, which is a different design.<Sampling Theory/Chapter 2: Simple Random Sampling>[Sampling Theory, Chapter 2, Simple Random Sampling, PDF pp. 1–4: replacement conventions and ordered versus unordered sample probabilities]{ref}
Every label has <inclusion probability> $n/N$, but equal <inclusion probabilities> alone are insufficient. For example, from six labels choose either $\{1,2,3\}$ or $\{4,5,6\}$ with <probability> $1/2$. Each label is included with <probability> $1/2$, yet eighteen of the twenty size-three <subsets> are impossible: this is a two-group <cluster sampling> design. <Unequal probability sampling> deliberately changes inclusion <probabilities>, often to improve precision for a particular <estimand>, and generally requires an appropriately weighted <estimator> rather than the plain <sample mean>.
Consider the original mathematical population with labelled values $(2,4,6,8,10,12)$, and let $n=2$. The fifteen equiprobable <subsets>, followed by their <sample means>, are
$$
\begin{array}{c|rrrrr}
\text{first value }2&\{2,4\}:3&\{2,6\}:4&\{2,8\}:5&\{2,10\}:6&\{2,12\}:7\\
4&\{4,6\}:5&\{4,8\}:6&\{4,10\}:7&\{4,12\}:8&\\
6&\{6,8\}:7&\{6,10\}:8&\{6,12\}:9&&\\
8&\{8,10\}:9&\{8,12\}:10&&&\\
10&\{10,12\}:11&&&&
\end{array}
$$
Their <sample means> sum to $105$, giving <expected value> $105/15=7$. Their squared deviations from $7$ sum to $70$, giving <variance> $70/15=14/3$. This is an enumeration over hypothetical selections from six stipulated numbers, not measured survey data.
\Image[/simple-random-sampling-subsets-and-replacement.png]
{title=All fifteen size-two subsets and all thirty-six ordered replacement outcomes from six labelled values}
With independent uniform <sampling with replacement>, the $6^2=36$ ordered outcomes each have <probability> $1/36$. The two orders $(2,4)$ and $(4,2)$ are distinct outcomes, and $(2,2)$ is now possible. Forgetting order does not make the resulting unordered multisets equiprobable: $\{2,4\}$ has <probability> $2/36$ but $\{2,2\}$ has <probability> $1/36$. The thirty-six <sample means> still have <expected value> $7$, but their squared deviations sum to $210$, giving <variance> $210/36=35/6$. Avoiding duplicate labels improves precision for this fixed population.
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