Suppose I tell you that my password is D!h98+3(dkE4. Do you conclude that since I don't want you to know my password, I must be trying to mislead you as to what my password is, and so the probability that this is my password is actually less than 1/95^12?

If I assert that the outcome as THHTT, either I'm lying or I'm not lying, and there's little evidence either way. What little evidence there is probably doesn't push my probability of telling the truth below 3%, and surely the strength of the evidence has little, if anything, to do with the prior probability of the coin showing THHTT.