 It’s natural to think we should use the more fine-grained division of possibilities, the three-way division in the case of Athena, Beatrice, and Cecil. But we can actually divide things further—infinitely further in fact. For example, Athena might win by a full length, by half a length, by a quarter of a length, etc. So the possibility that she wins is actually infinitely divisible. We can extend the PoI to handle such infinite divisions of possibilities in a natural way by saying that, if Athena wins, the probability that she’ll win by between 1 and 2 lengths is twice the probability that she’ll win by between \(1/2\) and 1 length. But the same problem we were trying to solve still persists, in the form of the notorious Bertrand paradox (Bertrand 2007 [1888]).