Residual properties of groups and probabilistic methods

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.