In this paper we will talk about conditions in which sequence of random variables converge, convergence rate and convergence probability. At the beginning we will recall some of the basic terms of probability such as independence of random variables, mathematic expectation and variance and Cebısev inequality. We will define few types of convergence of random variables, introduce the term of measure and measure integration and, comparing with them, we will state and prove Cebısev, Bernoulli and Khinchin weak law theorem of large numbers by using cutting method. We will be analyzing sequence of independent random variables that have ”zero – one” characteristic, Borel and Kolmogorov theorem and introduce term of tail functions and tail events. Furthermore, we will define empirical function of distribution, study convergence of series of random variables and prove Borel, Kolmogorov, Chung and Cantelli strong law of large numbers. In the end, we will bring up basic theorems needed to prove the law of repeated logarithm.