In this thesis, a brief introduction to probabilistic method is given. Probabilistic method is a powerful tool used to prove the existence of objects with certain properties. The key idea behind the method is the following: in order to show the existence of an object that has some wanted property, an adequate probability space is constructed and then we prove that a randomly chosen object in this space satisfies the wanted property with positive probability. In the first chapter, basic ideas of probabilistic method are showcased through a number of examples. The second chapter is dedicated to the concept of mathematical expectation which is widely applied in probabilistic method. We also consider the situations in which the method does not offer the solution immediately, but some alterations are necessary in order to get the end result. Third chapter introduces a very important theorem by Hungarian mathematician László Lovász - the local lemma. Local lemma is generally used when the objects of interest are scarce. Finally, the fourth chapter is dedicated to the use of concentration inequalities and martingales. Concentration inequalities are generally used to estimate the deviation of a random variable from its mean. Martingales are very useful tool of probabilistic method. Through some examples in graph theory, the use of martingales is demonstrated.