  The orbital parameters of extra-solar planets have a significant impact on
the probability that the planet will transit the host star. This was recently
demonstrated by the transit detection of HD 17156b whose favourable
eccentricity and argument of periastron dramatically increased its transit
likelihood. We present a study which provides a quantitative analysis of how
these two orbital parameters affect the geometric transit probability as a
function of period. Further, we apply these results to known radial velocity
planets and show that there are unexpectedly high transit probabilities for
planets at relatively long periods. For a photometric monitoring campaign which
aims to determine if the planet indeed transits, we calculate the expected
transiting planet yield and the significance of a potential null result, as
well as the subsequent constraints that may be applied to orbital parameters.
