The stochastic sequential assignment problem assigns distinct workers to sequentially arriving tasks with stochastic parameters. In this paper the assignments are performed so as to minimize the threshold probability, which is the probability of the long-run reward per task failing to achieve a target value (threshold). As the number of tasks approaches infinity, the problem is studied for independent and identically distributed (i.i.d.) tasks with a known distribution function and also for tasks that are derived from r distinct unobservable distributions (governed by a Markov chain). Stationary optimal policies are presented, which simultaneously minimize the threshold probability and achieve the optimal long-run expected reward per task.