Unmanned deep-sea and planetary vehicles operate in highly uncertain environments. Autonomous agents often are not adopted in these domains due to the risk of mission failure, and loss of vehicles. Prior work on contingent plan execution addresses this issue by placing bounds on uncertain variables and by providing consistency guarantees for a `worst-case' analysis, which tends to be too conservative for real-world applications. In this work, we unify features from trajectory optimization through risk-sensitive execution methods and high-level, contingent plan execution in order to extend existing guarantees of consistency for conditional plans to a chance-constrained setting. The result is a set of efficient algorithms for computing plan execution policies with explicit bounds on the risk of failure. To accomplish this, we introduce Probabilistic Temporal Plan Network (pTPN), which improve previous formulations, by incorporating probabilistic uncertainty and chance-constraints into the plan representation. We then introduce a novel method to the chance-constrained strong consistency problem, by leveraging a conflict-directed approach that searches for an execution policy that maximizes reward while meeting the risk constraint. Experimental results indicate that our approach for computing strongly consistent policies has an average scalability gain of about one order of magnitude, when compared to current methods based on chronological search.