If you don't even know what you mean by surprise (because that's what we're ostensibly trying to figure out, right?), then how can you use the math to deduce that some quantitative measure of surprise is equal in all cases?
Because the number of bits of information is the same in all cases. Any given random sequence provides evidence of countless extremely low probability world models - we just don't consider the vast majority of those world-models because they aren't elevated to our attention.
If in the future we want to keep using a coin flip example, I suggest forgetting things that are so mind-like as 'Alice is clairvoyant', and maybe just talk about biased and unbiased coins. It seems like an unnecessary complication.
It's both necessary and relevant. Indeed, I crafted my example to make your brain come up with that answer. Your conscious mind, once aware of it, probably immediately threw it into the "Silly explanation" column, and I'd hazard a guess that if asked, you'd say you wrote it down as a joke.
Because it clearly isn't an example of a world-model being allocated evidence. Your explanation is post-hoc - that is, you're rationalizing. Your description would be an elegant mathematical explanation - I just don't think it's correct, as pertains to what your mind is actually doing, and why you find some situations more surprising than others.
Because the number of bits of information is the same in all cases.
I don't know why you're using self-information/surprisal interchangeably with surprise. It's confusing.
Any given random sequence provides evidence of countless extremely low probability world models - we just don't consider the vast majority of those world-models because they aren't elevated to our attention.
Like in the sense that there are hypotheses that something omniscient would consider more likely conditional on Alice doing something surprising, that humans just don't think of ...
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.