And now probability

Though not exactly a follow-up on the earlier post, these thoughts came to me when writing it.

In Tales of Protection, Erik Fosnes Hansen bases his fiction on how a collection of seemingly random events are connected by a pattern. As an example:

The probability of heads or tails occurring when a coin is tossed is 0.5. If we get heads 9 times in a row, one would think that there is a greater chance of getting tails the next toss. Yet, the probability is still only 0.5 of getting heads (or tails). Over a large number of tosses, the heads and tails will even out according to its probability. This is fundamental in figuring probability and is call the Law of Large Numbers.

The paradox is that one toss of a coin is unrelated to the next, yet over a large number of tosses the number of heads attain an equilibrium value ( = 0.5 x number of tosses). How is this possible? If the two of us were tossing coins in different parts of the world, would heads turning up for me be affected by what you get?

Is this a self-referencing paradox related to the Gödel's Incompletenesss Theorem? On second thoughts, we probably should leave this much abused theorem be.