When we measure a source signal in the presence of a background rate that has been independently measured, the usual approach is to obtain an estimate of the background rate by observing an empty part of the sky, and an estimate of the source signal plus background rate by observing the region where a source signal is expected. The source signal rate is then estimated by subtracting the background rate from the source signal plus background rate. However, when the rates or their observation times are small, this procedure can lead to negative estimates of the source signal rate, even when it should produce a positive value. By applying the Bayesian approach, we solve the problem and prove that the most probable value of source signal rate is zero when the observed total count is smaller than the expected background counts. It is also shown that the results from the conventional method is consistent with the most probable value obtained from the Bayesian approach when the source signal is large or the observation time is long enough.