Predicting the number of positives resulting from testing a sample

The previous sections have been dedicated to the evaluation of the probability that a particular individual, tagged as positive in a test, is really infected. In those sections we have understood how, in absence of any other hints, it is important to know the percentage $p$ of infectees in the population. Knowing this parameter is paramount also for better designing a containing strategy in addressing the pandemic. Therefore we move now to the related, but quite different problem: `counting', although not in an exact way, the number of infected individuals in a population. Given the didactic spirit of this paper, we keep proceeding step-by-step. First we focus on the number of positives that we expect to observe if we check a sample using the quite imperfect test we are considering. Then we also take into account the effect of sampling a population, since, as it is rather obvious, the proportion of infected in a sample of size $n_s$ will not be exactly equal to that in the whole population of $N$ individuals. For this reason we distinguish, hereafter, $p_s$ of the sample from $p$ of the population.