But if you flip all heads, suddenly there are higher-probability alternatives. Not because all-heads is especially unlikely by chance, but because it's especially likely by not-chance. Maybe the coin is double-headed. Maybe it's weighted in some clever way[1]. Maybe you're hallucinating or dreaming. Maybe some god is having a laugh. All these things are (so at least it seems) much more likely to produce all-heads than a random-looking sequence.
Which is, I think, what is interesting about this: All-heads is no more improbable than any other random sequence, but in the case of an all-heads sequence, suddenly we start looking for laughing gods, hallucinations, or dreams as an explanation.
Which is to say, the interesting thing here is that we'd start looking for explanations of an all-heads sequence, even though it's no more improbable than any other sequence.
the interesting thing here is that we'd start looking for explanations of an all-heads sequence, even though it's no more improbable than any other sequence.
It's no more probable under the null hypothesis, but much more probable under more probable than average alternative hypotheses.
Alice: "I just flipped a coin [large number] times. Here's the sequence I got:
(Alice presents her sequence.)
Bob: No, you didn't. The probability of having gotten that particular sequence is 1/2^[large number]. Which is basically impossible. I don't believe you.
Alice: But I had to get some sequence or other. You'd make the same claim regardless of what sequence I showed you.
Bob: True. But am I really supposed to believe you that a 1/2^[large number] event happened, just because you tell me it did, or because you showed me a video of it happening, or even if I watched it happen with my own eyes? My observations are always fallible, and if you make an event improbable enough, why shouldn't I be skeptical even if I think I observed it?
Alice: Someone usually wins the lottery. Should the person who finds out that their ticket had the winning numbers believe the opposite, because winning is so improbable?
Bob: What's the difference between finding out you've won the lottery and finding out that your neighbor is a 500 year old vampire, or that your house is haunted by real ghosts? All of these events are extremely improbable given what we know of the world.
Alice: There's improbable, and then there's impossible. 500 year old vampires and ghosts don't exist.
Bob: As far as you know. And I bet more people claim to have seen ghosts than have won more than 100 million dollars in the lottery.
Alice: I still think there's something wrong with your reasoning here.