# Why does Fisher’s exact test use the hyper geometric distribution rather than the multi nomial distribution?

To understand hypergeometric and multinomial better, I’d like to know why fisher exact test used hypergeometric rather than multinomial distribution.

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.

• What’s a marginal?
– user271077
Feb 10 '20 at 15:04
• Answer edited to explain this. Feb 10 '20 at 15:06
• Marginal is a distribution here
– Nick
Feb 10 '20 at 15:06
• marginal distribution is the distribution of a single random variable alone Feb 10 '20 at 20:15
• Fisher's Exact Test is commonly used under a multinomial sampling scheme: conditioning on observed marginal totals isn't assuming they were fixed. See stats.stackexchange.com/q/441139/17230 & stats.stackexchange.com/q/364417/17230. Feb 16 '20 at 21:55