Upper Limits in the Case That Zero Events are Observed: An Intuitive Solution to the Background Dependence Puzzle

We compare the ``unified approach'' for the estimation of upper limits with an approach based on the Bayes theory, in the special case that no events are observed. The ``unified approach'' predicts, in this case, an upper limit that decreases with the increase in the expected level of background. This seems absurd. On the other hand, the Bayesian approach leads to a result which is background independent. An explanation of the Bayesian result is presented, together with suggested reasons for the paradoxical result of the ``unified approach''.