Survey for Transiting Extrasolar Planets in Stellar Systems: III. A Limit on the Fraction of Stars with Planets in the Open Cluster NGC 1245

We analyze a 19-night photometric search for transiting extrasolar planets in the open cluster NGC 1245. An automated transit search algorithm with quantitative selection criteria finds six transit candidates; none are bona fide planetary transits. We characterize the survey detection probability via Monte Carlo injection and recovery of realistic limb-darkened transits. We use this to derive upper limits on the fraction of cluster members with close-in Jupiter-radii, RJ, companions. We carefully analyze the random and systematic errors of the calculation. For similar photometric noise and weather properties as this survey, observing NGC 1245 twice as long results in a tighter constraint on "Hot Jupiter", HJ, companions than observing an additional cluster of similar richness as NGC 1245 for the same length of time as this survey. This survey observed ~870 cluster members. If 1% of stars have 1.5 RJ HJ companions, we expect to detect one planet for every 5000 dwarf stars observed for a month. To reach a ~2% upper limit on the fraction of stars with 1.5 RJ HJ companions, we conclude a total sample size of ~7400 dwarf stars observed for at least a month will be needed. Results for 1.0 RJ companions, without substantial improvement in the photometric precision, will require a small factor larger sample size.