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Suppose I have a population, $P$, of 100 individuals and I am devising 3 algorithmic methods to sample 20 people from said population: $A$, $B$, and $C$. We know that $C$ is a random, unbiased algorithm.

Suppose the outputs for each algorithm are 100-length lists of 0's and 1's. Binary data where a 1 indicated that an individual is selected for a sample. There are 20 1's in each list:

$A(P)$ = $[0, 1, 0, 1, ... , 1]$

$B(P)$ = $[1, 0, 1, 1, ... , 0]$

$C(P)$ = $[1, 0, 0, 1, ... , 1]$

If I know that C is random, but I suspect that $A$ is not random and that $B$ is random, how can I go about testing this? I initially tried McNemar's test due to having paired nominal data. But then realized the statistic would always be 0 since my each list contains exactly 20 1's. Is Fisher's Exact test the best test for this type of experiment? How can I test a suspicion that $B$ is likely random (or similar to $C$) and show that the opposite is true for $A$?

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    $\begingroup$ It's impossible to discern whether any sample is random based on this information. For instance, after obtaining sample $C,$ the experimenter might have numbered the population members so that the first 20 are in $C,$ whence $C(P) = (1,1,\ldots,1,0,\ldots,0).$ $\endgroup$ – whuber Jan 3 at 23:36
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Assuming, that the sequence of items is not important and has not been rearranged after one of the selection processes (see the comment of whuber), this sounds like a classical application of the Wald-Wolfowitz runs test, see Runs Test for Detecting Nonrandomness or Wald-Wolfowitz runs test. For example, in your case with $N_+=20$ and $N_-=80$ one gets a mean number of runs $\mu = 33$ with a standard deviation $\sigma = 3.17$, which can be used as basis for the hypothesis test. But please check, that this is based on the randomness / unbiasedness you are assuming.

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