How can you talk about it being +EV on subsequent flips if you think no probability exists for this situation? You don't know the EV if you don't know the probability. Why are you willing to hit the all-in button after 8 heads in a row? If your philosophy is consistent, then even after 8 heads in a row you still wouldn't know if P(heads) > 50%. For all you know, maybe the prior was so heavy on Tails that those 8 heads didn't sway the probability past 50%. So maybe Heads is a -EV bet.
But the thing is, we do know. When I say it's uniform I'm saying, "I know that I don't know," because uniform is synonymous with having zero info.
Quote:
If I told you I had an orange tree in my backyard and you could bet a dollar at even money on whether the oranges it grows are naturally orange or bright blue would you say it's 50/50?
Am I a human with life experience / knowledge of the world, or am I an alien from a distant galaxy? If I'm a human, then I have information and my prior distribution makes Orange a heavy favorite. If I'm an alien with no knowledge of Earth's fruits, then it's 1/(# of possible colors), so if you've already narrowed it down for me to Blue/Orange then it's 50/50.
Similarly, if you tell me two cricket teams are playing and ask me to pick the winner, my chance of guessing right is 50/50 because I have no clue about that sport. It doesn't matter if one of those teams is a 10:1 favorite. I know my chance is 50/50 even though I don't know the true chance of my team winning. Again, my claim of 50/50 amounts to, "I know that I don't know." 50/50 implies the maximum Shannon entropy, ie the least amount of information. (If Rusty is reading this, he can smack me on the head if I just shat on Shannon's grave.)
That sports example is probably something you can test via experiment if you don't believe me that it comes out to 50/50 (as opposed to, "there is no probability"). Better yet, a similar experiment can be done with a deck of cards. You can have someone remove cards from the deck and you have to guess red/black without knowing which cards they removed.