Or the coin being cheat, or some cheating or "non-random" effect in the situation. Delusional recollection of events.
How did I "rule out" the alternatives? When I imagine me doing that, I imagine me reasoning poorly. I go by Jaynes' policy of having a catch all "something I don't understand" hypothesis for multiple hypothesis testing. In this case, it would be "some agent action I can't detect or don't understand the mechanism of". How did I rule that out?
Suppose it's 1,000,000 coin flips, all heads. The probability of that is pretty damn low, and much much lower than my estimates for the alternatives, including the "something else" hypothesis. You can make some of that up with a sampling argument about all the "coin flip alternatives" one sees in a day, but that only takes you so far.
I don't see how I would ever be confident that 1,000,000 came up all heads with "fair" coin flipping.
The probability of that is pretty damn low
The probability of any specific sequence of 1M coin flips is "pretty damn low" in the same sense. The relevant thing here is not that that probability is low when they're all heads, but that the probability of some varieties of "something else" is very large, relative to that low probability. Or, more precisely, what sets us thinking of "something else" hypotheses is some (unknown) heuristic that tells us that it looks like the probability of "something else" should be muc...
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.