I feel a strong need to apologize for asking this question, so please pardon me for asking such a sloppy question wherein I ask you to refrain from formal proofs and answer in layman terms for such a topic.
Let's consider the Weak Law of Large Numbers (WLLN) for Bernoulli trials of '0' and '1', both with equal probability of happening. Now WLLN states that the average converges to 0.5, which is the expectation, for an infinite sequence or for a very large length of sequence.
I always have trouble with randomness, because if the experiment is truly random, and we let go of any faith that we have in structures of nature, we have a possibility, (irrespective of how low that possibility is) that the number "1" will occur repeatedly always for the entire sequence - "1111111111111111111111111111... up to infinite times".
There is no way of saying that, that sequence is not random.
In this case, the average is 1 and not 0.5. So, can we say that in this scenario, the weak law of large numbers did not hold true?
Or since the law of large numbers is "weak", it only works under the assumption that there is some structure in nature, or as Plato once imagined - the universe will attain a perfect state wherein we will have perfect patterns (like perfect circles) in nature, when the law of large numbers will always hold true?