Independent events in a simple random experiment and the meaning of independence

Abstract We count the number and patterns of pairs and tuples of independent events in the following simple random experiment: first a fair coin is flipped and then a fair die is tossed. The number of independent pairs equals 888,888, whereas the number of independent pairs for a biased coin and a biased die is the more “normal" looking number 124. We show also that the classic example of S. Bernstein of tossing a tetrahedron providing three pairwise independent, but not mutually independent events, is in a certain sense, unstable. This raises a question about the stability of independent events. These observations lead us to discuss two different concepts of independence. We also pose a few problems and hypotheses and give a few illustrative examples. However, in this short note we do not give a review of the vast literature related to these concepts.