Conclusions

The classical inverse problem related to the binomial distribution has been reviewed and extended to the presence of background either only on the number of `successes', or on the trials themselves. The probabilistic approach followed here allows to treat the problems only using probability rules. The results are always in qualitative agreement with intuition, are consistent with observations and prior knowledge and, never lead to absurdities, like $p$ outside the range 0 and 1.

The role of the priors, that are crucial to allow the probabilistic inversion and very useful to balance in the proper way prior knowledge and evidence from new observations, has been also emphasized, showing when they can be neglected and when they are so critical that it is preferable not to provide probabilistic conclusions.

It is a pleasure to thank Stefano Andreon for several stimulating discussions on the subject.