The commuting probability of a finite group is the probability that a
randomly chosen pair of elements of the group commute. Starting from Erdos and Turan (1968), many mathematicians have considered commuting probability and its generalizations over the years. Another popular topic, after the work of Brauer and Fowler (1955), is the study of graphs defined on groups motivated by commutativity. In this talk, we discuss various probabilities of finite groups (e.g.commuting probability, relative commuting probability, g-commuting probability, auto commuting probability, cyclicity degree etc.) and their relations with various graphs defined on groups (such as commuting graph, relative commuting graph, g-noncommuting graph, cyclic graph etc.).