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The sustainable medical supplier selection (SMSS) is an important issue facing the medical industry in the context of sustainable development, which can be regarded as a typical multi-attribute group decision making (MAGDM) problem. In the MAGDM process, linguistic term set (LTS) is particularly natural and convenient for decision makers (DMs) to express evaluation information. Especially, probabilistic linguistic term set (PLTS) is a very critical and effective tool, which can reflect the importance of different linguistic terms. Due to the different preferences and experience of different DMs, they may use multi-granularity probabilistic linguistic term sets (MGPLTSs) to represent different linguistic information. In this article, in order to study the comparison method of MGPLTSs, a new possibility degree formula is firstly proposed and its properties is proved. Then, in order to build a weight model, a possibility degree-based Best-Worst method (BWM) and a probability degree based-maximizing deviation method are established to calculate the subjective weights and objective weights of attributes, respectively. Where after, a MAGDM method is proposed by combining the ELimination Et Choix Traduisant la REalite (ELECTRE) method with Evaluation based on Distance from Average Solution (EDAS) method in the multi-granularity probabilistic linguistic information environment. Finally, the created MAGDM method is applied to the SMSS, and its effectiveness and advantages compared with other existing methods are verified.