  NASA's Kepler Space Telescope has successfully discovered thousands of
exoplanet candidates using the transit method, including hundreds of stars with
multiple transiting planets. In order to estimate the frequency of these
valuable systems, it is essential to account for the unique geometric
probabilities of detecting multiple transiting extrasolar planets around the
same parent star. In order to improve on previous studies that used numerical
methods, we have constructed an efficient, semi-analytical algorithm called
CORBITS which, given a collection of conjectured exoplanets orbiting a star,
computes the probability that any particular group of exoplanets can be
observed to transit. The algorithm applies theorems of elementary differential
geometry to compute the areas bounded by circular curves on the surface of a
sphere (see Ragozzine & Holman 2010). The implemented algorithm is more
accurate and orders of magnitude faster than previous algorithms, based on
comparisons with Monte Carlo simulations. We use CORBITS to show that the
present solar system would only show a maximum of 3 transiting planets, but
that this varies over time due to dynamical evolution. We also used CORBITS to
geometrically debias the period ratio and mutual Hill sphere distributions of
Kepler's multi-transiting planet candidates, which results in shifting these
distributions towards slightly larger values. In an Appendix, we present
additional semi-analytical methods for determining the frequency of exoplanet
mutual events, i.e., the geometric probability that two planets will transit
each other (Planet-Planet Occultation, relevant to transiting circumbinary
planets) and the probability that this transit occurs simultaneously as they
transit their star. The CORBITS algorithms and several worked examples are
publicly available at https://github.com/jbrakensiek/CORBITS
