yes you are 100% correct,the probability still remains same 1/2.
But as i understand,
The Law of Large Numbers states that the average of the results from multiple trials will tend to converge to its expected value (e.g. 0.5 in a coin toss experiment) as the sample size increases. The way I understand it, while the first 10 coin tosses may result in an average closer to 0 or 1 rather than 0.5, after 1000 tosses a statistician would expect the average to be very close to 0.5 and definitely 0.5 with an infinite number of trials.
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Image source(http://homepage.divms.uiowa.edu/~rdecook/stat1010/notes/Section_6.3_law_large_numbers.pdf)
yes the chance is 1/2 even after a billion trials are made,but i think we dont have to make infinte numbers of trials to get 1/2 result.we may acheive a closer value to 1/2 in million or billion flips.
Thats a finite number,my intuition told me that if that was case there might be boys only in one country and girls only in other country(small countries) but that is not happenning,the population is roughly even in finite 'n' trails i.e. 7.4 billion.
so i concluded that i should excpect or chance is more to get head in next trail in 1000tail/1000toss than 100tail/100toss.
I drawed the conclusion intuitionally,it might or might not be correct. But
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Indeed, the law of large numbers requires that the number of trials go to infinity. But any finite number is an infinite distance removed from infinity. Hence, even for a large amount of trials the law of large numbers cannot be applied. More generally, limiting behaviour is only valid in the limit and nowhere else.