Jumping to coincidences: Defying odds in the realm of the preposterous

James A. Hanley, Epidemiology and Biostatistics, McGill The calculation of probabilities is central to statistical inferences; however, researchers, especially those who are not trained in probability, often have difficulty (or err) when setting up correct probability calculations. Probabilities of seemingly rare events that are assessed after the fact are especially problematic. In an early report, I described three examples where probability specialists themselves have been "near-sighted" in assessing or predicting usual and unusual events generated by state lotteries. In one example, a lottery official offered "data" which one should expect from a fair lottery; unfortunately, the logic used to "predict the usual" was faulty. In the two others, the unusual (what one should not often expect) was calculated to be much more unusual than it really was. Since then, the story of a fourth -- and seemingly very unusual -- lottery event, and the official statistical reaction to it, were carried in worldwide publications. Again, the reported reaction was based on faulty probability calculations. These four faulty lottery calculations, and several new ones involving unusual births and birthdays, prompted me to bring together in one place my "case series" and share it with a larger audience. I argue that these variants of a common "probability blind spot" are not sufficiently appreciated and that we need to be very skeptical of after-the-fact probability calculations. These miscalculations have serious implications for the interpretation of high-throughput or high-dimensional data, and indeed for the interpretation of the scientific literature in general.