One bit of the analysis is left out (if you follow the link to the Math Stack Exchange post) -- the probability of a bin being empty is not independent of the p

One bit of the analysis is left out (if you follow the link to the Math Stack Exchange post) -- the probability of a bin being empty is not independent of the probability of other bins being empty. From the perspective that the number of groupings Y(x) is contingent on the total number of bins filled, X, you might say they're assuming E[Y(X)] = E[Y(E[X])]. But as n -> infinity, the effect of this becomes negligible.