  In this paper we define for a Bernoulli samples the \emph{ empirical
infection measure}, which counts the number of positives (infections) in the
Bernoulli sample and for the \emph{ pool samples} we define the empirical pool
infection measure, which counts the number of positive (infected) pool samples.
For this empirical measures we prove a joint large deviation principle for
Bernoulli samples. We also found an asymptotic relationship between the \emph{
proportion of infected individuals } with respect to the samples size, $n$ and
the \emph{ proportion of infected pool samples} with respect to the number of
pool samples, $k(n).$ All rate functions are expressed in terms of relative
entropies.
