The great interest aroused by the incorporation of Statistics and Probability into curricular projects has been accompanied by considerable evidence of significant difficulties in the meaningful learning and application of the concepts. These difficulties have been the subject of many studies, mostly concerning secondary school students. This study seeks to investigate the level of understanding of random phenomena, in a group of university students, on the second year of a Technical and Industrial Engineering Degree at the University of the Basque Country, in Spain. The students, who had all undertaken an introductory course in Probability Theory, were tested through both written questionnaires and one-to-one interviews, where they were presented with situations that demanded the ability to apply the knowledge they had learnt, at a high level, to a range of situations and to justify their reasoning. The results show that the vast majority of students had a poor understanding of random phenomena, applying alternative ideas to the ones taught on their course in Probability. The need to improve teaching strategies in Probability and Statistics, so that students can develop skills in constructing probabilistic models, is highlighted. (Contains 7 tables.)