  Several experiments in high-energy physics and astrophysics can be treated as
on/off measurements, where an observation potentially containing a new source
or effect ("on" measurement) is contrasted with a background-only observation
free of the effect ("off" measurement). In counting experiments, the
significance of the new source or effect can be estimated with a widely-used
formula from [LiMa], which assumes that both measurements are Poisson random
variables. In this paper we study three other cases: i) the ideal case where
the background measurement has no uncertainty, which can be used to study the
maximum sensitivity that an instrument can achieve, ii) the case where the
background estimate $b$ in the off measurement has an additional systematic
uncertainty, and iii) the case where $b$ is a Gaussian random variable instead
of a Poisson random variable. The latter case applies when $b$ comes from a
model fitted on archival or ancillary data, or from the interpolation of a
function fitted on data surrounding the candidate new source/effect.
Practitioners typically use in this case a formula which is only valid when $b$
is large and when its uncertainty is very small, while we derive a general
formula that can be applied in all regimes. We also develop simple methods that
can be used to assess how much an estimate of significance is sensitive to
systematic uncertainties on the efficiency or on the background. Examples of
applications include the detection of short Gamma-Ray Bursts and of new X-ray
or $\gamma$-ray sources.
