  Many physical phenomena are modeled as stochastic searchers looking for
targets. In these models, the probability that a searcher finds a particular
target, its so-called hitting probability, is often of considerable interest.
In this work we determine hitting probabilities for stochastic search processes
conditioned on being faster than a random short time. Such times have been used
to model stochastic resetting or stochastic inactivation. These results apply
to any search process, diffusive or otherwise, whose unconditional short-time
behavior can be adequately approximated, which we characterize for broad
classes of stochastic search. We illustrate these results in several examples
and show that the conditional hitting probabilities depend predominantly on the
relative geodesic lengths between the initial position of the searcher and the
targets. Finally, we apply these results to a canonical evidence accumulation
model for decision making.
