Do extraordinary claims require extraordinary evidence?

No.

If the slogan were true, we could never have sufficient evidence for a whole host of extraordinarily improbable events.

Although the slogan is catchy, it fails to appreciate all of the factors needed to asses the probability that an event occurred.

Most importantly, the likelihood that if the extraordinary event had not occurred, what’s the probability that we’d have the evidence that we do?

For example, consider a pick in the lottery, for which the odds are a 300 million to one. If the slogan were true, the evidence presented by the lottery commission announcing the winning pick would be swamped by the improbability that reported pick was in fact the winning number.

But the evidence that it takes to counterbalance the low probability of a person’s winning lottery pick needn’t be enormous or unusual at all. It’s just needs to be more probable given the truth of the hypothesis than its falsehood.

In the lottery context, what’s the likelihood that the lottery commission would’ve announced a particular number if it weren’t the winning pick?

If that probability is sufficiently low, it can counterbalance any intrinsic improbability in the number itself.

In the election context, we ask what’s the likelihood that in all contested states, we’d have dozens of eyewitnesses attesting fraud and numerous statistical regularities strongly indicative of fraud, if election fraud (extraordinary event) did not occur?

If the probability is sufficiently low that we would not have dozens of witnesses to fraud and numerous statistical indicators of fraud unless fraud had occurred, then evidence of fraud can counterbalance the intrinsic improbability of a stolen election.