In industrial-sized cyber-physical systems, ensuring fulfillment of requirements gets increasingly more costly as the number of components increases. To make the task feasible, compositional verification has been suggested as a scalable solution. Such techniques allow verification by divide-and-conquer, often using assume-guarantee contracts. Although previous research has focused mostly on the non-probabilistic setting, in the real world, probabilities often arise due to random hardware failures, stochastic communication delays, sensor ghost objects, machine learning components, rounding errors caused by finite-precision arithmetic, human behavior, and probabilistic algorithms. Therefore, for contract theories to be practically relevant to cyber-physical systems, there is a need to support probabilistic reasoning, for instance regarding safety and reliability. To this end, we propose a completely trace-based probabilistic contract theory, supporting general probability measures, continuous time, and continuous state spaces. To verify decompositions of such contracts, we also present a deductive system, which is illustrated on an industrially inspired automatic emergency braking example.