  For decades researchers have studied the On/Off counting problem, where a
measured rate consists of two parts. One due to a signal process and another
due to a background process, of which both magnitudes are unknown. While most
frequentist methods are adequate for large count numbers, they cannot be
applied to sparse data. Here I want to present a new objective Bayesian
solution that only depends on three parameters: the number of events in the
signal region, the number of events in the background region, and the ratio of
the exposure for both regions. First, the probability of the hypothesis that
the counts are due to background only is derived analytically. Second, the
marginalized posterior for the signal parameter is also derived analytically.
With this two-step approach it is easy to calculate the signal's significance,
strength, uncertainty, or upper limit in a unified way. The approach is valid
without restrictions for any count number including zero and may be widely
applied in particle physics, cosmic-ray physics and high-energy astrophysics.
In order to demonstrate its performance I apply the method to gamma-ray burst
data.
