  The experimental problem of converting a measured binomial quantity, the
fraction of events in a sample that pass a cut, into a physical binomial
quantity, the fraction of events originating from a signal source, is described
as a system of linear equations. This linear system illustrates several
familiar aspects of experimental data analysis. Bayesian probability theory is
used to find a solution to this binomial measurement problem that allows for
the straightforward construction of confidence intervals. This solution is also
shown to provide an unbiased formalism for evaluating the behavior of data sets
under different choices of cuts, including a cut designed to increase the
significance of a possible, albeit previously unseen, signal.
  Several examples are used to illustrate the features of this method,
including the discovery of the top quark and searches for new particles
produced in association with $W^{\pm}$ bosons. It is also demonstrated how to
use this method to make projections for the potential discovery of a Standard
Model Higgs boson at a Tevatron Run 2 experiment, as well as the utility of
measuring the integrated luminosity through inclusive $p\bar{p} \to W^{\pm}$
production.
