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The following project is in the field of combinatorics. It is a broad summary of the probabilistic method. Basic definitions and theorems, needed for the proper use of the probabilistic method, are introduced. Simple examples are provided in order to illustrate the theory in practice and to introduce the method itself. The project focuses on competitive problems from major national and international competitions, in whose solutions could be used probabilistic method, contributing to their briefness. One of our main aims is demonstrating the broad application of the probabilistic method not only in some theoretical fields of mathematics, but also in competitive problems. The problems in the abstract are with increased difficulty and we believe, they can serve as a preparation for mathematical olympiads and competititons.