Expectation of 500 coin flips after 500 realizations

$ \begingroup $ I was hoping person could provide clearness surrounding the comply scenario. You are asked “ What is the expected issue of observe heads and tails if you flip a bonny coin 1000 times ”. Knowing that coin flips are i.i.d. events, and relying on the law of big numbers you calculate it to be :
$ $ N_ { heads } = 500 \ ; N_ { tails } = 500 $ $
now, let us have observed/realized the inaugural 500 flips to all be heads. We want to know the update expect total of realizations of the remaining 500 flips. Because the first 500 events have been realized and they do not effect the underlying physical coin flipping process, we know that the have a bun in the oven count of heads and tails of the remaining 500 flips are :

$ $ N_ { heads } = 250 \ ; N_ { tails } = 250 $ $
so, here is my question/confusion : I understand that each coin flick is freelancer and that any unmarried individual coin throw has a probability of $ \frac { 1 } { 2 } $ coming up heads. however, based on the police of boastfully numbers we know that the ( if we value tails as 0 and heads as 1 ) entail of the tosses will approach $ 0.5 $ as the act of tosses approaches $ \infty $. So, based on that, if we have observed 500 heads in a row, why do we not statistically expect to realize more tails going forward ? I fully realize the follow think is incorrect, but it feels like we are ( statistically ) due for a tails and that the probability of tails should be raised and heads lowered. Since this is not the case, it feels as though this is conflicting with the original anticipation of $ N_ { heads } = 500 $ and $ N_ { tails } = 500 $.

Again, I realize that this intelligent is wrong, but I am hoping person can help me understand why this past information ( 500 realizations of heads in a rowing ) does not provide any newfangled, update information that updates the probability for the remaining flips ? distinctly the coin does not know that it barely came up heads $ 500 $ times, so is the compensate way to think about this that the law of bombastic numbers does n’t imply that in the following 500 flips tails is more probably, but quite that as $ N \rightarrow \infty $ we expect 50 % of realizations to be heads and 50 % to be tails. In which font my error in reasoning is based on applying a limit theorem that applies in the asymptote to a preasymptotic position ?

I besides feel like this has to deal with a bite of confusion between individual events ( a individual coin flip coming up heads ), and the collective action of a stage set of events ( 1000 mint tosses ) that expose nonrandom properties. After searching I came across a fantastic quote by Kolmogorov $ ^1 $ :

“ In world, however, the epistemic prize of the theory of probability is revealed entirely by limit theorems. … In fact, all epistemic measure of the theory of probability is based on this : that large-scale random phenomena in their collective action create stern, nonrandom regularity. The very concept of mathematical probability would be bootless if it did not find its realization in the frequency of happening of events under large-scale repetition and uniform conditions. ”

I believe this quote clears up some of my confusion, but if anyone could elaborate on why realizations ( based on a sleep together statistical serve ) can not be used to update subsequent probabilities, I would greatly appreciate it !

1. B. V. Gnedenko and A. N. Kolmogorov: Limit distributions for sums of independent random variables. Addison-Wesley Mathematics Series

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