In safety-critical deep learning applications robustness measurement is a vital pre-deployment phase. However, existing robustness verification methods are not sufficiently practical for deploying machine learning systems in the real world. On the one hand, these methods attempt to claim that no perturbations can ``fool'' deep neural networks (DNNs), which may be too stringent in practice. On the other hand, existing works rigorously consider $L_p$ bounded additive perturbations on the pixel space, although perturbations, such as colour shifting and geometric transformations, are more practically and frequently occurring in the real world. Thus, from the practical standpoint, we present a novel and general {\it probabilistic robustness assessment method} (PRoA) based on the adaptive concentration, and it can measure the robustness of deep learning models against functional perturbations. PRoA can provide statistical guarantees on the probabilistic robustness of a model, \textit{i.e.}, the probability of failure encountered by the trained model after deployment. Our experiments demonstrate the effectiveness and flexibility of PRoA in terms of evaluating the probabilistic robustness against a broad range of functional perturbations, and PRoA can scale well to various large-scale deep neural networks compared to existing state-of-the-art baselines. For the purpose of reproducibility, we release our tool on GitHub: \url{ https://github.com/TrustAI/PRoA}.