We outline a probabilistic approach to the solution of a number of problems
concerning residual properties of infinite groups. This approach gives ris
e to a new and short proof of a well known conjecture of Magnus. It also yi
elds new results on residual proper-ties of the modular group PSL2(Z) and o
f other free products of finite groups. Similar ideas can be used to show t
he existence of finitely generated dense free subgroups in various profinit
e groups, and yield an analogue of the Tits alternative for profinite compl
etions of linear groups. (C) 2001 Academie des sciences/Editions scientifiq
ues et medicales Elsevier SAS.