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  "body": "\n## Definition\n\nConsider a $2 \\times 2$ [[contingency table|PANSMHE0SW]] of counts $a, b$ in the first row and $c, d$ in the second row, with total $n = a + b + c + d$.\nFix the row totals $r_1 = a + b$, $r_2 = c + d$ and the column totals $k_1 = a + c$, $k_2 = b + d$.\nAssume the null hypothesis that the row factor and the column factor are independent.\nThen, conditionally on these totals, $a$ has the [[hypergeometric distribution|0SPBPW3K61]]\n$$P(a) = \\binom{r_1}{a} \\binom{r_2}{k_1 - a} \\Big/ \\binom{n}{k_1}.$$\n*Fisher's exact test* computes the [[p-value|0EHYZFZ4PK]] as a sum of $P(a')$ over tables with the same totals.\nThe sum runs over the tables at least as extreme as the observed one.\nThe one-sided version sums over $a' \\ge a$; the usual two-sided version sums over all $a'$ with $P(a') \\le P(a)$.\n\n## Remarks\n\nExample: in the lady tasting tea experiment, $r_1 = r_2 = k_1 = k_2 = 4$, $n = 8$, and she classifies all cups correctly, so $a = 4$.\nThen $P(4) = \\binom{4}{4}\\binom{4}{0}/\\binom{8}{4} = 1/70$, and the one-sided p-value is $1/70 \\approx 0.014$.\nNon-example: replacing the hypergeometric law by its chi-squared approximation gives Pearson's [[chi-squared test|0EN0V3NY20]], whose p-value is only approximate.\nThe test is conservative: since the conditional law is discrete, the probability of a type I error is at most $\\alpha$ and often smaller.\n",
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