Condition monitoring and diagnostics of a combined cycle gas turbine (CCGT) power plant has become an important tool to improve its availability, reliability, and performance. However, there are two major challenges in the diagnostics of performance degradation and anomaly in a single-shaft combined cycle (CC) power plant. First, since the gas turbine (GT) and steam turbine (ST) in such a plant share a common generator, each turbine's contribution to the total plant power output is not directly measured, but must be accurately estimated to identify the possible causes of plant level degradation. Second, multivariate operational data instrumented from a power plant need to be used in the plant model calibration, power splitting, and degradation diagnostics. Sensor data always contain some degree of uncertainty. This adds to the difficulty of both estimation of GT to ST power split (PS) and degradation diagnostics. This paper presents an integrated probabilistic methodology for accurate power splitting and the degradation diagnostics of a single-shaft CC plant, accounting for uncertainties in the measured data. The method integrates the Bayesian inference approach, thermodynamic physics modeling, and sensed operational data seamlessly. The physics-based thermodynamic heat balance model is first established to model the power plant components and their thermodynamic relationships. The model is calibrated to model the plant performance at the design conditions of its main components. The calibrated model is then employed to simulate the plant performance at various operating conditions. A Bayesian inference method is next developed to determine the PS between the GT and the ST by comparing the measured and expected power outputs at different operation conditions, considering uncertainties in multiple measured variables. The calibrated model and calculated PS are further applied to pinpoint the possible causes at individual components resulting in the plant level degradation. The proposed methodology is demonstrated using operational data from a real-world single-shaft CC power plant with a known degradation issue. This study provides an effective probabilistic methodology to accurately split the power for degradation diagnostics of a single-shaft CC plant, addressing the uncertainties in multiple measured variables.