According to the so-called "Birthday Paradox", only 23 people need be chosen at random for there to be a 50:50 chance that at least two of those selected will share the same birthday. In this paper, I generalise this argument to the case of N people being chosen at random, such that there are 50:50 odds of at least two belong to one of G groups (where, for example, G = 365 for birthdays, 12 for star-signs etc). It turns out that the number of people needed to give such odds is given by N = 1.2sqrt(G) - in other words, the number scales non-linearly with G. We hypothesize that the reason people are surprised by the low number predicted by this formula is that people tend to use a simple linear scaling law to guesstimate the number of people needed, that is N(guess) = k(G). This would have the effect of making guesses ever more unreliable, the larger G becomes (and thus the greater the "outlandishness" of the coincidence). We suggest that the use of a linear law to approximate a non-linear phenomenon may help explain why so many feel the need to reach for "spooky" explanations of perfectly explicable coincidences. (A subsequent paper , in which Susan Blackmore and I carried out experimental tests, tends to confirm this hypothesis)