Although this article discusses problem solving in the context of an introductory probability course, the discussion of the kinds of errors students make, and the methods the author uses to improve students' problem solving skills could be applied, with small changes, to a variety of introductory courses which have a substantial problem-solving component. However, they tend to use these words loosely, and in the process, fall prey to fallacies and errors. The mathematical notion of probability is a different case.
Terms are well-defined, rules formulated and proven, reason preferred over intuition.
Yet because of this dual nature of probability, students often find the topic difficult, especially when concepts such as conditional probability, Bayes' rule and their ilk are thrown in. I have been handling an introductory probability course for engineering and mathematics juniors for five years. Apparently, many of them do not have a solid conceptual understanding of basic probability concepts, and resort to mindless substitution in formulas and frantic number crunching instead. Once they understand the problem, they can then use the rules and theorems they have learned to tackle it well. I decided to train my class in those two strategies.