Estimate the probabilities of finding two messages with the same MD5 checksum, given total numbers of messages of 263, 264, and 265. And again the probability that the k+ 1th message has a different checksum from each of the preceding kis 1-k/2 128
However, the approximation in the hint there for simplifying the product fails rather badly now. So, instead, take the log of each side and use the approximation log(1 - k/2 128) ≈ -k/2 128 .