When solving group decision-making (GDM) problems, the proper values of consensus thresholds are vital for consensus checking and improving. Decision makers are usually not confident to select a proper threshold. In the linguistic setting, this paper presents a new process to support the determination of consensus thresholds. A new concept, namely random consensus index, is presented to serve as a reference object so that the risk preference of a decision maker could be induced by comparing the quality of the information in hands with the averaging quality of randomly generated information. In this sense, admissible consensus thresholds could be determined automatically, and decision makers are not necessary to focus on the detail of consensus measures. Furthermore, the process is applied to the GDM problems with probabilistic linguistic preference relations, and a new GDM approach is presented. Different from the existing contributions, the consensus is measured by the probability of a set of preference relations being with acceptable consensus, and then the consensus is improved by revising the involved probability distributions. All the involved parameters are intuitive and interpretable. A case study, regarding the departure audit in China, demonstrates that the proposed GDM approach is effective even if the original preferences of experts are with low consistency and consensus.