Comment by dragonwriter

There will always be improbable sequences; with a fair coin, every possible sequence of length N is equally improbable, after all; if you flip a fair coin 64 times, the sequence is guaranteed to be a 1 in 2^64 event.

OTOH, the probability of some other explanation besides a fair coin isn’t consistent among all other possible sequences, so what the actual result does to your estimate of the likelihood of a fair coin depends on the actual sequence, and your basis for believing the coin was fair going in.

Things are only slightly different with, say, a coin you’ve been told has a 60% bias.

EDIT: For instance, if there is a 1:1,000,000 chance that you would be given an underestimate of bias and a 1:1,000,000,000 chance of the outcome you actually receive being true if the coin had only the bias you were informed of, its a lot more likely that you were lied to than that you just got an unusually consistent set of results.