Teasers, paradoxes and fallacies in stochastics

Theory, Research & Practice in Mathematical Education
Bentz, H.
Bell, A., Low, B., & Kilpatrick, J.
Shell Centre for Mathematical Education
Nottingham, UK

While solving stochastical problems one often notices a certain discrepancy between the intuitive reasoning of the person involved, and the "objective" causes given by the mathematical theory. So, the paths to follow in either direction will usually turn out to be different ones and will not always lead to the same final answer.We have the greatest difficulties to grasp the origins and effects of chance and randomness. Also, the history of probability reports some problems and paradoxical examples which support the suspicion that stochastics is a rather exceptional science even within the mathematical fields. I shall introduce and discuss a small collection of problems of that kind i.e. problems which carry certain counterintuitive aspects. My objections here are manifold. First of all, the discussion of such problems, especially in the classroom, helps (i) to clarify ambiguous stochastical situations, (ii) to understand basic concepts on this field, (iii) to interpret formulations and results. Then, we, the teachers and professionals, can use them to test our own intuitive level of understanding. Finally, as those "paradoxes" and teasers have an entertaining aspect too, we should make use of this to increase the motivation of the students occasionally. Six of these problems were chosen to be discussed in the sequel. Here, I have tried to present them in a unique form. First, the problem will be formulated, an illustration included. Then, a "hint" is given, which adds (or stresses) some information about hidden processes or about strategies, which I recommend to follow. Thirdly, one solution is outlined, although very often several different approaches are known. Where possible I have chosen the one which follows a general idea. Finally, some variations, comments and references are added.