Summing up

The simple reasoning based on mean expectations leads to correct results only when all probabilistic effects are negligible, an approximation which holds, generally speaking, only for `large numbers'. Under this approximation the numbers of individuals tagged as Positive or Negative can be considered to follow in a deterministic way from the assumptions, one of which is the proportion of infectees. This number can then be obtained inverting the deterministic relation, thus yielding Eq. (). But when fluctuations around the mean expectations become important we need to use probability theory in order to infer the parameter of interest.

As far as telling from a single test if a person tagged as Positive is really infected, we have seen that the prior `assumed proportion' of infected individuals in the entire populations plays a major role. We have seen how to get the probability of interest reasoning on the fraction of positives really infected in the sample of positives. In more general terms this probability has to be calculated using Bayes' theorem, that will be shortly reminded in the next section.