This paper tackles the challenge of learning multi-goal dexterous hand manipulation tasks using model-based Reinforcement Learning. We propose Goal-Conditioned Probabilistic Model Predictive Control (GC-PMPC) by designing probabilistic neural network ensembles to describe the high-dimensional dexterous hand dynamics and introducing an asynchronous MPC policy to meet the control frequency requirements in real-world dexterous hand systems. Extensive evaluations on four simulated Shadow Hand manipulation scenarios with randomly generated goals demonstrate GC-PMPC's superior performance over state-of-the-art baselines. It successfully drives a cable-driven Dexterous hand, DexHand 021 with 12 Active DOFs and 5 tactile sensors, to learn manipulating a cubic die to three goal poses within approximately 80 minutes of interactions, demonstrating exceptional learning efficiency and control performance on a cost-effective dexterous hand platform.