When a new counting experiment is proposed, it is crucial to predict whether the desired source signal will be detected, or how much observation time is required in order to detect the signal at a certain significance level. The concept of the a priori prediction of the detection limit in a newly proposed experiment should be distinguished from the a posteriori claim or decision whether a source signal was detected in an experiment already performed, and the calculation of statistical significance of a measured source signal. We formulate precise definitions of these concepts based on the statistical theory of hypothesis testing, and derive an approximate formula to estimate quickly the a priori detection limit of expected Poissonian source signals. A more accurate algorithm for calculating the detection limits in a counting experiment is also proposed. The formula and the proposed algorithm may be used for the estimation of required integration or observation time in proposals of new experiments. Applications include the calculation of integration time required for the detection of faint emission lines in a newly proposed spectroscopic observation, and the detection of faint sources in a new imaging observation. We apply the results to the calculation of observation time required to claim the detection of the surface thermal emission from neutron stars with two virtual instruments.