Pretending for the sake of argument that I don't see any regularities in your sequence that I wouldn't expect from genuinely random coin flips (it actually looks to me more human-generated, but with only n=36 I'm not very confident of that): the odds are pretty much the same as the prior odds that you'd actually flip a coin 36 times rather than just writing down random-looking Hs and Ts.
You said something interesting there, and then skipped right past it. That's the substance of the question. You don't get to ignore those regularities; they do, in fact, affect the probabilities. Saying that they don't appear in the ratio of Bayes' Rule is... well, misusing Bayes' Rule to discard meaningful evidence.
The only formula I wrote down was "2^-n times Pr(Alice cheated)" and those probabilities are definitely not greater than 1. Would you care to be more explicit?
2^(-n) approaches 1 as n approaches infinity, but for any finite n, is greater than 1. Multiply that by a probability of 1, and you get a probability greater than 1. [ETA: Gyah. It's been too long since I've done exponents (literally, ten years since I've done anything interesting). You're right, I'm confusing negative exponents with division in exponents.]
Saying that they don't occur in the ratio of Bayes' Rule [...]
But I didn't say that. I didn't say anything even slightly like that.
This is at least partly my fault because I was too lazy to write everything out explicitly. Let me do so now; perhaps it will clarify. Suppose X is some long random-looking sequence of n heads and tails.
Odds(Alice cheated : Alice flipped honestly | result was X) = Odds(Alice cheated : Alice flipped honestly) . Odds(result was X | Alice cheated : Alice flipped honestly).
The second factor on the RHS is, switching from my eccen...
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.