  We introduce a likelihood analysis of multi-epoch stellar line-of-sight
velocities to constrain the binary fractions and binary period distributions of
dwarf spheroidal galaxies. This method is applied to multi-epoch data from the
Magellan/MMFS survey of the Carina, Fornax, Sculptor and Sextans dSph galaxies,
after applying a model for the measurement errors that accounts for binary
orbital motion. We find that the Fornax, Sculptor, and Sextans dSphs are
consistent with having binary populations similar to that of Milky Way field
binaries to within 68% confidence limits, whereas the Carina dSph is remarkably
deficient in binaries with periods less than ~10 years. If Carina is assumed to
have a period distribution identical to that of the Milky Way field, its
best-fit binary fraction is 0.14^{+0.28}_{-0.05}, and is constrained to be less
than 0.5 at the 90% confidence level; thus it is unlikely to host a binary
population identical to that of the Milky Way field. By contrast, the best-fit
binary fraction of the combined sample of all four galaxies is
0.46^{+0.13}_{-0.09}, consistent with that of Milky Way field binaries. More
generally, we infer probability distributions in binary fraction, mean orbital
period, and dispersion of periods for each galaxy in the sample. Looking ahead
to future surveys, we show that the allowed parameter space of binary fraction
and period distribution parameters in dSphs will be narrowed significantly by a
large multi-epoch survey. However, there is a degeneracy between the parameters
that is unlikely to be broken unless the measurement error is of order ~0.1
km/s or smaller, presently attainable only by a high-resolution spectrograph.
