To understand hypergeometric and multinomial better, I’d like to know why fisher exact test used hypergeometric rather than multinomial distribution.
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Fisher's exact test treats the marginals as fixed, so you are dealing with a problem of the kind "draw $n_{11}$ observations (first row first column cell entry) of out $n_{1.}$ (first row marginal) without replacement with overall $n$ observations of which $n_{.1}$ are in the first column marginal". By "first row/column marginal" I mean the total number of observations in the first row/column.
For the multinomial you'd have probabilities for all cells, but marginals could not be fixed.
Christian Hennig
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What’s a marginal? – Feb 10 '20 at 15:04
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Answer edited to explain this. – Christian Hennig Feb 10 '20 at 15:06
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Marginal is a distribution here – Nick Feb 10 '20 at 15:06
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marginal distribution is the distribution of a single random variable alone – GENIVI-LEARNER Feb 10 '20 at 20:15
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Fisher's Exact Test is commonly used under a multinomial sampling scheme: conditioning on observed marginal totals isn't assuming they were fixed. See https://stats.stackexchange.com/q/441139/17230 & https://stats.stackexchange.com/q/364417/17230. – Scortchi - Reinstate Monica Feb 16 '20 at 21:55
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Fair enough, edited answer. – Christian Hennig Feb 16 '20 at 23:15