A well known problem in counting experiments is that we are rarely in the ideal situation of being able to count individually and at a given time all the objects of interest. More often we have to rely an a sample of them. Other problems that occur in real environments, especially in frontier research, are detector inefficiency and presence of background: sometimes we lose objects in counting; other times we might be confused by other objects that do not belong to the classes we are looking for, though they are observationally indistinguishable from the objects of interest.
We focus here on the effect of background in measurements of proportions. For a extensive treatment of the effect of background on rates, i.e. measuring the intensity of a Poisson process in presence of background, see Ref. , as well as chapters 7 and 13 of Ref. .
The paper is structured as follows. In section 2 we introduce the `direct' and `inverse' probabilistic problems related to the binomial distribution and the two cases of background that will be considered. In section 3 we go through the standard text-book case in which background is absent, but we discuss also, in some depth, the issue of how prior knowledge does or does not influence the probabilistic conclusions. Then, in the following two sections we come to the specific issue of this paper, and finally the paper ends with the customary short conclusions.