Large numbers, Large spaces, Bernoulli

This equation ( and as promised the latex is in the title=""; block ) is described as
The average (overbar X_n) is the sum (Σ (X₁....X_n) of a sequence as it tends to infinity (∞) divided into equal partitions (1/n) to its extension(n). It is referenced at wiki as The Law of Large Numbers.

The reason I am referencing this is the fact that I was considering probability and specifically the divergence of probability from its norm. This is a statement of exactly what I had deduced, without an extension that I considered. The derivative of the probability and the divergence in sampling. It should be obvious that at small numbers of n (like 1) a coin toss is completely at odds ( pun intended ) with its probability, as it is either heads or tails and as such 100% on the sampling.

What I was considering was a room filled with pennies and shoveling those pennies into a truck. It would seem to be a reasonable conclusion that the pennies were virtually 50:50 in heads and tails. I was wondering what I might think if my first shovel full were all heads and would I be correct in assuming that it would converge to 50:50, even though it would seem that assuming an even distribution in the remaining coins would be just as reasonable, and I think this verges on the Bob Barker problem of selection sets. This:, If I pick a card at random, and then somebody takes the remaining cards in the deck and removes all but the aces, would you re-choose?, and of course you should.

A reasonable assumption of probability can be changed by subsequent action or information. In this case I cannot be certain that somebody has not played a trick on me and filled a room with pennies that are only heads, but I would assume that my premise is correct, though in contrast to a literal interpretation, that the probability of probability p' of tails is increased in the remaining coins.

I could be completely off my mark on this, but it is tempting me to do a test. When I am dealing with random numbers and its application to security, I may use this as a game to play while I create and test the code. I know that it is a fallacy in the short term and the concept would not give any advantage to a gambler, however it interests me as it applies to very large numbers that must be equal, like conservation of momentum and thus equal reaction. I have seen and made conjecture about anti-matter in the past, and there is an issue there if all things are equal ( as all information and measurement indicates ), a left handed universe ( all matter, no anti-matter) with conservation is very odd.