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In the last months, due to the emergency of Covid-19, questions related tothe fact of belonging or not to a particular class of individuals (`infected ornot infected'), after being tagged as `positive' or `negative' by a test, havenever been so popular. Similarly, there has been strong interest in estimatingthe proportion of a population expected to hold a given characteristics(`having or having had the virus'). Taking the cue from the many relateddiscussions on the media, in addition to those to which we took part, weanalyze these questions from a probabilistic perspective (`Bayesian'),considering several effects that play a role in evaluating the probabilities ofinterest. The resulting paper, written with didactic intent, is rather generaland not strictly related to pandemics: the basic ideas of Bayesian inferenceare introduced and the uncertainties on the performances of the tests aretreated using the metrological concepts of `systematics', and are propagatedinto the quantities of interest following the rules of probability theory; theseparation of `statistical' and `systematic' contributions to the uncertaintyon the inferred proportion of infectees allows to optimize the sample size; therole of `priors', often overlooked, is stressed, however recommending the useof `flat priors', since the resulting posterior distribution can be `reshaped'by an `informative prior' in a later step; details on the calculations aregiven, also deriving useful approximated formulae, the tough work being howeverdone with the help of direct Monte Carlo simulations and Markov Chain MonteCarlo, implemented in R and JAGS (relevant code provided in appendix).

Checking individuals and sampling populations with imperfect tests