The Chance Probability of the Quadrupole-Octopole Alignment: A Monte Carlo Census of Random Axes

The quadrupole and octopole of the microwave sky define preferred axes, and the two axes are reported to lie within roughly ten degrees of each other, a coincidence often cited as evidence against statistical isotropy. This paper computes how surprising that coincidence is, and no more than that: two million pairs of random unsigned axes give a probability of 0.01537 for a separation of ten degrees or less, about one chance in 65.1, in agreement with the closed-form value of 0.01519. The alignment is therefore unusual but far from impossible on a single sky, and when a second coincidence of the kind commonly bundled with it is required jointly, the probability falls to 0.00053. The honest reading is that the alignment is a two-to-three-sigma class event whose evidential weight depends entirely on how many other coincidences were available to be noticed, a dependence this paper quantifies. The study is simulated, not measured.