The probability fuzzy soft set (FSS) is an approach that is obtained by adding a probability degree to each approximate element of the FSS and is formed by combining fuzzy set theory and soft set theory. This idea was later studied in different sets, such as intuitionistic fuzzy sets, neutrosophic fuzzy sets, Pythagorean fuzzy sets, etc. In this paper, a new possibility set is defined in the Fermatean fuzzy environment, and the essential features of new sets have been examined. The set-theoretic operations of new possibility sets, such as subset, soft equal, complement, union, intersection, AND, and OR, have been characterized with the help of elaborated examples. Their fundamental laws and properties are also discussed. A practical example concerning quality assurance in distance education is studied to demonstrate the applicability of new similarity measures (SMs) in decision-making situations. Thus, it has been shown that the newly given SM can be successfully applied to real-world decision problems. Finally, a comparison of the similarity of the proposed model is made with some existing models.