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 rise to a new and short proof of a well known conjecture of Magnus. It also yields new results on residual properties of the modular group PSL₂(Z) and of other free products of finite groups. Similar ideas can be used to show the existence of finitely generated dense free subgroups in various profinite groups, and yield an analogue of the Tits alternative for profinite completions of linear groups.