Let G be a finite group and let ν(G) be the probability that two randomly selected elements of G produce a nilpotent group. In this article we show that for every positive integer n>0, there is a finite group G such that ν(G)=1n. We also classify all groups G with ν(G)=12. Further, we prove that if G is a solvable nonnilpotent group of even order, then ν(G)≤p+34p, where p is the smallest odd prime divisor of |G|, and that equality exists if and only if GZ∞(G) is isomorphic to the dihedral group of order 2p where Z∞(G) is the hypercenter of G. Finally we find an upper bound for ν(G) in terms of |G| where G ranges over all groups of odd square-free order.

Keywords: Soluble group, Nilpotent subgroup, Probability