  Many field stars reside in binaries, and the analysis and interpretation of
photometric and spectroscopic surveys must take this into account. We have
developed a model to predict how binaries influence the scientific results
inferred from large spectroscopic surveys. Based on the rapid binary evolution
code BSE, it allows us to model a representative population of binaries and
generate synthetic survey observations. We describe this model in detail, and
apply it to the radial velocity variation of subgiant and giant stars in the
Galactic disc, as observed by the Apache Point Observatory Galactic Evolution
Experiment (APOGEE), part of the Sloan Digital Sky Survey III. APOGEE provides
an excellent data set for testing our binary models since a large fraction of
the stars have been observed multiple times.
  By comparing our model to the APOGEE observations we constrain the initial
binary fraction of solar-metallicity stars in the sample to be $f_{\rm
b,0}=0.35\pm0.01$, in line with the solar neighbourhood. We find that the
binary fraction is higher at lower metallicities, consistent with other
observational studies. Our model matches the shape of the high-velocity scatter
in APOGEE, which suggests that most velocity variability above 0.5 km/s comes
from binaries. Our exploration of binary initial properties shows that APOGEE
is mostly sensitive to binaries with periods between 3 and 3000 years, and is
largely insensitive to the detailed properties of the population. We can,
however, rule out a population where the mass of the lower-mass star is drawn
from the IMF independently of its companion.
