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Cross-classify the binary exposure $A_i=1_{\{Z_i\ge 1\}}$ against the binary outcome $Y_i$. The four cells count placebo survivors or deaths and active-treatment survivors or deaths.

Under the sharp causal null hypothesis, every observed outcome equals the fixed value $Y_i(0)=Y_i(1)=Y_i(2)$, regardless of assignment. With fixed arm sizes, the active set is a uniformly chosen subset of size $n_1+n_2$. Under independent assignment, $A_i$ are independent Bernoulli variables with probability ${\pi }_1+{\pi }_2$, and conditioning on their total again makes the active set uniform. Conditional on the table margins, the active-group outcome count therefore has exactly the hypergeometric distribution used by Fisher's exact test.

The conditional p-value is super-uniform for every set of margins. The law of total probability then givesHence the test is valid under either assignment mechanism.

For a two-by-two contingency table, Fisher's exact test conditions on both row totals and both column totals. Under the null of no association, the upper-left count $K$ then has the hypergeometric distributionwhere $r_1$ and $c_1$ are the first row and column totals. A one-sided p-value sums the appropriate hypergeometric tail; a common two-sided p-value sums the probabilities of all feasible tables no more likely under the null than the observed table. Because the conditional distribution is discrete, the p-value is super-uniform random variable rather than generally exactly uniform.