In this talk, we will give a brief survey of some aspects in the probability group theory. We will also focus on two old problems given by Erdos and Turan [1] on random generating set for a given group G, and Miller [2] on random commuting pair of elements in a finite group G. More precisely, for every finite group G we would like to find the probability that the group generated by given two randomly chosen elements $x$ and $y$ to be a whole group. Moreover, we may state the similar probability when two elements of G are chosen randomly and independently commute. There are some known results given by some authors which will be given in the present talk and also state some recent researches in this area.

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