Let G be a finite group, then the probability that a randomly chosen pair of elements of G commutes is defined to be the number of pairs (x, y) 2 G×G = G2 with xy = yx divided to |G|2. Two usual ways to generalize this probability is to consider n-tuples (x1, x2, . . . , xn) of elements in group G with the property that xixj = xjxi for all 1  i, j  n or n + 1-tuples (x1, x2, . . . , xn+1) such that [x1, x2, . . . , xn+1] = 1. We denote them by PrnCom(G) and d(n)(G), and are called the probability of mutually commuting n-tuples and n-th nilpotency degree of G, respectively. In this talk, we will give some results concerning the above two probabilities and state some relations between these and the notion of the isoclinism. Furthermore, we give one more way of generalization of the probability for infinite groups at the end. 0

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