Consider two populations of zeros and ones. Let p₁ be the proportion of ones in the first population, and let p₂ be the proportion of ones in the second population. We would like to test the null hypothesis that p₁ = p₂ on the basis of a simple random sample from each population. Let n₁ be the size of the sample from population 1, and let n₂ be the size of the sample from population 2. Let G be the total number of ones in both samples. If the null hypothesis be true, the two samples are like one larger sample from a single population of zeros and ones. The allocation of ones between the two samples would be expected to be proportional to the relative sizes of the samples, but would have some chance variability. Conditional on G and the two sample sizes, under the null hypothesis, the tickets in the first sample are like a random sample of size n₁ without replacement from a collection of N = n₁ + n₂ units of which G are labeled with ones. Thus, under the null hypothesis, the number of tickets labeled with ones in the first sample has (conditional on G) an hypergeometric distribution with parameters N, G, and n₁. Fisher's exact test uses this distribution to set the ranges of observed values of the number of ones in the first sample for which we would reject the null hypothesis.