Edwards wrongly replaces k^3 by k^2 in his derivation, but this does not matter since k cancels out in numerator & denominator below. (So do the factors of 2 and pi.) When Edwards integrates, one way to try to do it (not his) is to compute int(r=0..a-x) / int(r=0..infinity) another way is to compute int(x=0..a-r) / int(x=0..infinity). My point is, probability = a ratio, numerator counts ways to be happy, denominator counts all ways. I'm not necessarily claiming either of these two ways is actually valid. Anyhow, the latter is what Edwards does, except he OMITS the denominator! Also, Edwards makes an implicit assumption that the tuple (P, x) is uniformly distributed within the outer circle. This is nowhere justified and is in fact wrong... since... the denominator he omits obviously has to have some nonobvious nonuniform probability distribution for x, to avoid infinite nonsense results. So: Edwards made plenty of errors, his derivation is garbage.