The Kepler mission is uniquely suited to study the frequencies of extrasolar planets. This goal requires knowledge of the incidence of false positives such as eclipsing binaries in the background of the targets, or physically bound to them, which can mimic the photometric signal of a transiting planet. Using real noise level estimates, we compute the number and characteristics of detectable eclipsing pairs involving main-sequence stars and non-main-sequence stars or planets, and we quantify the fraction of those that would pass the Kepler candidate vetting procedure. By comparing their distribution with that of the Kepler Objects of Interest (KOIs) detected during the first six quarters of operation of the spacecraft, we infer the false positive rate of Kepler and study its dependence on spectral type, candidate planet size, and orbital period. In the process, we also derive a prescription for the signal recovery rate of Kepler that enables a good match to both the KOI size and orbital period distribution, as well as their signal-to-noise distribution.