This paper describes a classroom teaching experiment, concerning the concept of probability, with children aged 5 in a kindergarten school. The teaching experiment was based on constructivist and interactionist theories about the learning of school mathematics and lasted one month. The collection of the information was based on the tape-recorded interviews with the children (each child was interviewed prior to the research program, at the end of the program and one month later) and the videotaped teaching sessions. During the program, we identified three critical steps in the development of the children's probabilistic thinking: a) the interpretation of the "different" outcomes in a two stage experiment, b) the acceptance of the realization of the experiment for resolving their conflicting viewpoints, and c) estimating the outcomes in a problem. At the end of the program the majority of the children managed to overcome their subjective interpretations and seemed to develop a primitive quantitative reasoning in probabilistic tasks. (Contains 2 diagrams.)