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In many of the answers and comments on the well-known XKCD #936: Short complex password, or long dictionary passphrase? question, the importance was stressed of generating the password randomly, and not just making one up. In particular, the current second highest voted answer on that question says:
Random choices are random and uniform. This is hard to achieve with human users. You must convince them to use a device for good randomness (a coin, not a brain), and to accept the result. This is the gist of my original answer (reproduced below). If the users alter the choices, if only by generating another password if the one they got "does not please them", then they depart from random uniformity, and the entropy can only be lowered (maximum entropy is achieved with uniform randomness; you cannot get better, but you can get much worse).
This got me wondering: what if I don't accept the result and instead generate a new random password? How big of an impact would that have on the entropy of the final generated password?
Presumably the more times I allow myself to reject a password, the more the entropy of the final password could potentially be reduced, so let's say for example that I generate 8 passwords and pick the one I like best. How much could that potentially reduce the effective entropy of the password I select? (Maybe
log_2(8) = 3 bits as a worst-case scenario?)