= Simple random sample
In finite-population sampling, a <simple random sample> of fixed size $n$ without replacement is a subset $S$ of $N$ labelled units drawn so that every size-$n$ subset has probability
$$
\mathbb P(S=s)=\binom Nn^{-1}.
$$
The sample is the selected subset; <simple random sampling> is the procedure that generates it. This definition does not require the observed values to be distinct, only the selected population labels. The without-replacement convention is important because an independent repeated draw from a population is a different design.<Unequal Probability Sampling in Penn State STAT 506>[Lesson 3, simple-random-sampling specialization]{ref}
One construction randomly orders all $N$ labels and takes the first $n$. Each particular subset can occupy those positions in $n!(N-n)!$ permutations, out of $N!$, proving the probability above. Equivalently, draw successively and uniformly from the remaining labels, then discard their order.
For $N=5,n=2$, there are $\binom52=10$ possible samples. A particular unit belongs to four of them, so its <inclusion probability> is $2/5$; a particular pair is one complete sample, so its <joint inclusion probability> is $1/10$. These are properties of the procedure before observing which subset was obtained.
A <simple random sample> need not resemble the whole population in every realization. For fixed values $(0,0,0,0,100)$, six of the ten size-two samples contain only zeros. Thus the <sample mean> is zero with probability $0.6$, although its <expected value> is the <finite population mean> $20$. The remaining four samples each have mean $50$, giving $0.6\times0+0.4\times50=20$. Unbiasedness concerns repeated selection, rather than a guarantee about any one sample. Coverage of the population labels and faithful implementation of the selection rule remain essential.
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