An important question for the European Union is to know whether its institutions will permit it to escape from political deadlocks each time a question is at stake. Two studies [Baldwin, Felsenthal1] suggest, by using the Impartial Culture assumption to model the voting behavior, that the EU could only take a decision in 2% of the cases with its current voting mechanisms. The Impartial Culture model has been criticized from a theoretical point of view [Feix], a political one [Moberg] and does not fit with experimental data [Gelman]. The generalized Impartial Anonymous Culture assumption we consider in this paper is an improvement of this first model. We here study the probability of approval under this assumption, first for the asymptotic case (reached when the number of countries $N$ goes to infinity), and next with computer enumerations and Monte Carlo simulations for the European Union with 27 members. We consider both the Treaty of Nice and some proposals for the European Constitution.