A very natural notion of equivalence of programs is the observational equivalence: two programs are equivalent if they have the same behaviour in any environment. More specifically in the lambda-calculus two terms M and N are equivalent if for every context C, the terms C[M] and C[N] are either both solvable or both unsolvable. Respecting this equivalence is an important property of some models of the lambda-calculus, called the full abstraction. It is well known that the model of infinitely extensional Böhm trees is fully abstract. The question we are interested in is what becomes of these notions when we introduce probabilistic choice in the calculus.

The solvability has a natural extension, which is the convergence probability, hence two terms are observationally equivalent if they have the same convergence probability under any context. There is also a simple way to define probabilistic Bohm trees, as we can associate to any term a subprobability distribution over head normal forms. But it is not obvious that these generalizations of the deterministic notions are still equivalent. In this talk we will show how to prove that probabilistic Böhm trees actually form a model, and how to get a separability result to ensure this model is fully abstract.