  Wide-field surveys for transiting planets, such as the NASA Kepler and TESS
missions, are usually conducted without knowing which stars have binary
companions. Unresolved and unrecognized binaries give rise to systematic errors
in planet occurrence rates, including misclassified planets and mistakes in
completeness corrections. The individual errors can have different signs,
making it difficult to anticipate the net effect on inferred occurrence rates.
Here we use simplified models of signal-to-noise limited transit surveys to try
and clarify the situation. We derive a formula for the apparent occurrence rate
density measured by an observer who falsely assumes all stars are single. The
formula depends on the binary fraction; the mass function of the secondary
stars; and the true occurrence of planets around primaries, secondaries, and
single stars. It also takes into account the Malmquist bias by which binaries
are over-represented in flux-limited samples. Application of the formula to an
idealized Kepler-like survey shows that for planets larger than 2 $R_\oplus$,
the net systematic error is of order 5%. In particular, unrecognized binaries
are unlikely to be the reason for the apparent discrepancies between hot
Jupiter occurrence rates measured in different surveys. For smaller planets the
errors are potentially larger: the occurrence of Earth-sized planets could be
overestimated by as much as 50%. We also show that whenever high-resolution
imaging reveals a transit host star to be a binary, the planet is usually more
likely to orbit the primary star than the secondary star.
