Exact power of Fisher's exact test (${\sum }_{(b,c)\in \mathcal{R}}{\mathrm{Bin}}_{n,p_0}(b){\mathrm{Bin}}_{n,p_1}(c)$)

For two independent binomial counts, form the Fisher exact test rejection region from their conditional null hypergeometric distribution. Summing alternative joint binomial probabilities over that region gives its exact power. For equal group sizes, the null allocation is symmetric, so the two-sided probability-ordering test agrees with doubled lower-tail probabilities. Conservative test discreteness can make a normal-approximation sample size underpowered even when both samples are large but event counts are low.