We define the empirical positives measure as a stochastic measure that quantifies the number of infections in a Bernoulli sample. Similarly, we define the empirical pool measure as a stochastic measure that enumerates the number of infected pool samples. A joint large deviation principle (LDP) is established for both empirical measures derived from the Bernoulli samples. Using this joint LDP, we establish a correlation between the asymptotic ratio of infected individuals to the sample size and the asymptotic ratio of infected pool samples to the total number of pool samples.