In a project, a mission completion time for each mission should be presented as stochastic due to uncertainly available machines/human resources. Besides, a charge for each mission is affected according to the completion time of the mission. In order to describe the uncertain completion time and charges, a stochastic completion time project (SCTP) is formulated in this paper. Then, SCTP reliability is defined as the probability that total completion time and budget constraints can be satisfied when completing a project. According to the previous reference, maximal and minimal capacity vectors bind all the feasible capacity vectors. Specifically, a previous reference calculated an estimated SCTP reliability in terms of an interval. On the other hand, we develop a breakdown approach such that all the feasible capacity vectors can be decomposed. Then, a recursive function is constructed based on the sum of disjoint products (SDP) principle. An algorithm is further presented to calculate the exact SCTP reliability. Project supervisors and teams can analyze and manage their projects in terms of SCTP reliability.