We study finite automata running over infinite binary trees and we relax the notion of accepting run by allowing a negligible set (in the sense of measure theory) of non-accepting branches. In this qualitative setting, a tree is accepted by the automaton if there exists a run over this tree in which almost every branch is accepting. This leads to a new class of tree languages, called the qualitative tree languages that enjoys many properties. Then, we replace the existential quantification -- a tree is accepted if there exists some accepting run over the input tree -- by a probabilistic quantification -- a tree is accepted if almost every run over the input tree is accepting. Together with the qualitative acceptance and the Buchi condition, we obtain a class of probabilistic tree automata with a decidable emptiness problem. To our knowledge, this is the first positive result for a class of probabilistic automaton over infinite trees.