Have to say I'm more warm towards a Hitchens razor than a Sagan standard.
The Sagan standard is actually based directly on probability reasoning.
Say you're trying to screen people for a disease. You have a rather good test that has 90% sensitivity (true-positive rate) and a 9% false-positive rate, and somebody tests positive. How probable is it that they have the disease? Answer: we have no idea because we missed out the
prior probability. In this case, how probable it is that some random person has the disease before we do the test, i.e. obtain the
evidence?
If the prevalence in the population is low, say 1%, then we have a problem because the answer to how likely is it that our positive test indicates the disease is a mere 9%. Basically all the false-positives dominate the result.
An extraordinary claim must be assigned a very low prior probability—that's what extraordinary
means. Hence, you need very, very good evidence that has very high sensitivity and almost no chance of a false positive, i.e extraordinary evidence.