  Stars are born from dense cores in molecular clouds. Observationally, it is
crucial to capture the formation of cores in order to understand the necessary
conditions and rate of the star formation process. The {\it Atacama Large
Mm/sub-mm Array} (ALMA) is extremely powerful for identifying dense gas
structures, including cores, at mm wavelengths via their dust continuum
emission. Here we use ALMA to carry out a survey of dense gas and cores in the
central region of the massive ($\sim10^5\:M_\odot$) Infrared Dark Cloud (IRDC)
G28.37+0.07. The observation consists of a mosaic of 86 pointings of the
12m-array and produces an unprecedented view of the densest structures of this
IRDC. In this first paper about this data set, we focus on a comparison between
the 1.3 mm continuum emission and a mid-infrared (MIR) extinction map of the
IRDC. This allows estimation of the "dense gas" detection probability function
(DPF), i.e., as a function of the local mass surface density, $\Sigma$, for
various choices of thresholds of mm continuum emission to define "dense gas".
We then estimate the dense gas mass fraction, $f_{\rm dg}$, in the central
region of the IRDC and, via extrapolation with the DPF and the known $\Sigma$
probability distribution function, to the larger-scale surrounding regions,
finding values of about 5\% to 15\% for the fiducial choice of threshold. We
argue that this observed dense gas is a good tracer of the protostellar core
population and, in this context, estimate a star formation efficiency per
free-fall time in the central IRDC region of $\epsilon_{\rm ff}\sim$10\%, with
approximately a factor of two systematic uncertainties.
