We obtain a surprisingly explicit formula for the number of random elements needed to generate a finite d-generator group with high probability. As a corollary we prove that if G is a d-generated linear group of dimension n then cd+logn random generators suffice.
Changing perspective we investigate profinite groups F which can be generated by a bounded number of elements with positive probability. In response to a question of Shalev we characterize such groups in terms of certain finite quotients with a transparent structure. As a consequence we settle several problems of Lucchini, Lubotzky, Mann and Segal.
As a byproduct of our techniques we obtain that the number of r-relator groups of order n is at most ncr as conjectured by Mann.