To my understanding:

Flipping a coin has a discrete 1/2 probability to be heads or tails, and every iteration of that trial resets the probability back to 1/2. So, it could be heads every time, or, heads 20% of the time. Both of these outcomes are perfectly valid. Yet, the more you do the experiment, the more the results tend to normalize into 50% heads and 50% tails as you approach infinity.

So my question is: what governs these outcomes and causes distributions to equalize/find equilibrium?

Edit: To clarify, I am not looking for mathematical proofs or theorems, I am simply looking for intuition regarding the ontology of this matter as it applies to our reality and the law of large numbers. I'm simply demonstrating my question in the context of a coin toss experiment.

  • 6
    $\begingroup$ This is the law of large numbers. $\endgroup$
    – Dave
    Jul 13, 2021 at 19:36
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    $\begingroup$ @spyter Is it fair to say that you're asking for intuition about how the law-of-large-numbers comes about, rather than a restatement of the proof of the theorem? Note that "every iteration of that trial resets the probability back to 1/2" describes independence. $\endgroup$
    – Sycorax
    Jul 13, 2021 at 19:46
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    $\begingroup$ This "equalization" is reminiscent of a common misunderstanding. The counts of heads and tails do not tend to equality. Quite the contrary: they tend to diverge. The equalization is that of relative frequencies. Ultimately this is a postulate, rather than a fact about the universe, because the universe has no way to test an asymptotic proposition. Far before any asymptotics kick in, the "coin" will wear down or otherwise physically change its characteristics. This is why commenters are right to insist on the purely mathematical nature of this phenomenon. $\endgroup$
    – whuber
    Jul 13, 2021 at 21:27
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    $\begingroup$ You seem to be missing how "randomness" works across sums of variates and hence averages. Let $X_i$ be $1$ if the $i$th toss is a head and $0$ otherwise and let the coin be fair & the trials independent. While the multivariate distribution of $X_1, X_2 ... X_L$ (for some large value $L$, say) is uniform over $\{0,1\}^L$ (which is what I assume you mean by "random"), when you sum those $L$ variates, there are vastly more ways to get a sum close to $L/2$ than there are close to $0$ or $L$, i.e. many more combinations that yield a proportion near $1/2$. As $L$ increases this effect increases... $\endgroup$
    – Glen_b
    Jul 14, 2021 at 1:46
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    $\begingroup$ ... and the proportion-of-heads distribution become more concentrated around $1/2$. You see this already start at $L=2$; there's two ways to get 50% heads but only one way to get 0% or 100% heads. $\endgroup$
    – Glen_b
    Jul 14, 2021 at 1:48


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