  In 2006, Varacca and V\"olzer proved that on finite graphs, omega-regular
large sets coincide with omega-regular sets of probability 1, by using the
existence of positional strategies in the related Banach-Mazur games. Motivated
by this result, we try to understand relations between sets of probability 1
and various notions of simple strategies (including those introduced in a
recent paper of Gr\"adel and Lessenich). Then, we introduce a generalisation of
the classical Banach-Mazur game and in particular, a probabilistic version
whose goal is to characterise sets of probability 1 (as classical Banach-Mazur
games characterise large sets). We obtain a determinacy result for these games,
when the winning set is a countable intersection of open sets.
