  We use a large set of radial velocities in the Ursa Minor and Draco dwarf
spheroidal galaxies to search for binary stars and to infer the binary
frequency. Of the 118 stars in our sample with multiple observations, six are
velocity variables with $\chi^2$ probabilities below 0.001. We use Monte Carlo
simulations that mimic our observations to determine the efficiency with which
our observations find binary stars. Our best, though significantly uncertain,
estimate of the binary frequency for stars near the turnoff in Draco and UMi is
0.2--0.3 per decade of period in the vicinity of periods of one year, which is
3--5$\times$ that found for the solar neighborhood. This frequency is high
enough that binary stars might significantly affect the measured velocity
dispersions of some dwarf spheroidal galaxies according to some previous
numerical experiments. However, in the course of performing our own
experiments, we discovered that this previous work had inadvertently
overestimated binary orbital velocities. Our first set of simulations of the
effects of binaries is based on the observed scatter in the individual velocity
measurements for the multiply-observed Draco and Ursa Minor stars. This scatter
is small compared to measured velocity dispersions and, so, the effect of
binaries on the dispersions is slight. This result is supported by our second
set of experiments, which are based on a model binary population normalized by
the observed binary frequency in Draco and Ursa Minor. We conclude that binary
stars have had no significant effect on the measured velocity dispersion and
inferred mass-to-light ratio of any dwarf spheroidal galaxy.
