How do you know how many Heads and Tails the coin has flipped in its lifetime? Maybe its previous owner flipped a bunch of Heads. Maybe it flipped some Heads while it was jangling in your pocket. And if you don’t know, how the hell do you expect the coin to know?!
If that doesn't help your intuition, maybe try this technically incorrect view. Those 'extra' Heads you feel are due can appear at any point in the entirety of the future. What are the chances they show up in the next five minutes, never mind the next billion years? You're looking for a handful events spread over the entire rest of time itself, after all.
This is wrong. It takes considerably fewer flips than "infinity" to see distinct convergence here.
If the coin is truly even - you have a binomial distribution, and the probability that you are far out to either edge of that distribution (nearly all heads, or nearly all tails) is vanishingly small with just 1000 flips.
Without any such guarantee of the larger result, all you have is a 50/50 chance for the next flip. With infinity, you can pick out a portion of the sequence with an unusual number of Tails and expect there to be an offsetting number of Heads elsewhere in the full (infinite) sequence.
As you mentioned, you don't see the convergence to 50% with any finite sample size, because that behavior is only guaranteed at infinity. All this to say, infinities are weird. 10,000,000 + infinity exactly equals infinity.
A random number <=0.5 is heads, >0.5 is tails.
Flip 100 million coins.
After you get 10 of the same result in a row, count how many times the 11th is the same as the first 10.
The odds of getting 12 Heads in a row is quite low.
The odds of getting 12 Heads in a row when you’ve already observed 11 coin flips land heads up is 50%.
Knowing that the odds of 12 Heads in a row is low, that you’ve seen 11 Heads in a row, and that - if you observe the next flip land heads up - you’ll have observed something quite rare, doesn’t influence the next coin flip.
It doesn't make sense because the events are independent. It's not a better or worse bet to pick one or the other. It doesn't make sense to think one is more than 50% likely because of a series of other independent events when you know the odds a priori.
This is incorrect. The convergence is to: (current heads + 0.5(n flips)/
(current flips + n flips).
For a gambler flipping coins, If they are neutral at the time of prediction, you expect them to end neutral.
If they are up flips when you enter the room, you expect them to the run up flips.
The odds of getting the sequence HTTHT are exactly the same as the odds of getting HHHHH, and similarly the odds of getting HTTHTH are exactly the same as HHHHHH.
Another way of thinking of it is that there's roughly a one in a thousand chance of getting 10 heads in a row; there's a one in two-thousand chance of getting 11 heads in a row, but if you've already gotten 10 heads in a row, you've already ruled out the ~1998/2000[1] possibilities that can't ever result in 11 heads (because they didn't start out with 10 heads in a row), so you only need to rule out one of the ~2/2000 remaining possibilities.
1: Actual numbers is 2046/2048 and 2/2048
If you knew, a priori, that 75% of the next 4 flips will land on H, and you've seen THH, you are justified in guessing H. In this scenario, H was actually due.
I think it's clear that real coins make no such promises.
And yes, I realize that after n flips of Heads, the probability of the n + 1th flip being Heads or Tails is equal. i.e. P(Heads) ^ n * P(Heads) = P(Heads) ^ n * P(Tails), because P(Heads) = P(Tails) = 0.5