Before revision:  The simplest case is when each obstacle is randomly chosen from two possible line segments. We define the problem SEGMENT COVER capturing the situation. In SEGMENT COVER, we are given a set of uncertain segments, each uncertain segment defined to be a random interval, chosen from two possible sub-intervals of the unit interval. Then, SEGMENT COVER asks if the probability of covering the entire unit interval is non-zero. We prove that SEGMENT COVER and hence CONTIGUOUS SAT are NP-hard.
After revision:  Segment cover is equivalent to a restricted version of SAT which we call contiguous SAT. Consider a SAT with the following restrictions. An input formula is in CNF form and an ordering of the clauses is given in which clauses containing any fixed literal appear contiguously. We call this restricted problem contiguous SAT. Our main result is that the problems segment cover and contiguous SAT are NP-hard.