Persi Diaconis gave the second annual Billingsley Lecture at UChicago yesterday on the topic of coincidences and what a skeptical statistician/probabilist should say about them. He started out by talking about how Jung was fascinated by paradoxes (apparently there’s one about having fish come up all the time in conversation).
It was mostly a general-audience talk (with some asides about Poisson approximation), and the first part on the birthday problem and variants. Abstracted away, the question is given balls (people) and bins/categories (days), how big should be so that there’s an even chance that two balls land in the same bin? Turns out , as we know, but we can expand this to deal with approximate matches (you need only 7 people to get 2 birthdays in the same week with probability around 1/2). If you want to put a graph on it you can ask social-network coincidence questions and get some scalings as a function of the number of edges and number of categories — here there are vertices and colors for the vertices. What these calculations show, of course, is that most coincidences are not so surprising, at least in this probabilistic sense. Some more advanced treatment might be found in Sukhada Fadnavis’s preprint (which also has something about a “shameful conjecture” on chromatic polynomials that was proved in 2000, but I don’t know why it is shameful). The second part of the talk was on problems arising in the study of ESP — namely that experimental controls are not really present, so the notion of a “trial” is hard to pin down, leading (of course) to more false perceptions of coincidences are being surprising. He closed with some remarks about how our perception of coincidence is really about how our minds work, and pointed to some work by Ruma Falk for those who are interested in that angle of things.
I was unaware of this body of Diaconis’s work, and it was nice to have a high-level talk to cap off the day.