  We report new methods for evaluating realistic observing programs that search
stars for planets by direct imaging, where observations are selected from an
optimized star list, and where stars can be observed multiple times. We show
how these methods bring critical insight into the design of the mission & its
instruments. These methods provide an estimate of the outcome of the observing
program: the probability distribution of discoveries (detection and/or
characterization), & an estimate of the occurrence rate of planets (eta). We
show that these parameters can be accurately estimated from a single mission
simulation, without the need for a complete Monte Carlo mission simulation, &
we prove the accuracy of this new approach. Our methods provide the tools to
define a mission for a particular science goal, for example defined by the
expected number of discoveries and its confidence level. We detail how an
optimized star list can be built & how successive observations can be selected.
Our approach also provides other critical mission attributes, such as the
number of stars expected to be searched, & the probability of zero discoveries.
Because these attributes depend strongly on the mission scale, our methods are
directly applicable to the design of such future missions & provide guidance to
the mission & instrument design based on scientific performance. We illustrate
our new methods with practical calculations & exploratory design reference
missions for JWST operating with a distant starshade to reduce scattered and
diffracted starlight on the focal plane. We estimate that 5 habitable
Earth-mass planets would be discovered & characterized with spectroscopy, with
a probability of 0 discoveries of 0.004, assuming a small fraction of JWST
observing time (7%), eta=0.3, and 70 observing visits, limited by starshade
fuel.
