A handful of coin flips is chaos; a million is clockwork. The law of large numbers is why averages are trustworthy even though each flip is pure chance.
Law of Large NumbersLive
chance averages out
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Flip a coin a few times and the proportion of heads jumps around wildly; flip it thousands of times and it settles onto the true probability. That is the law of large numbers — averages converge even though individual flips stay random. Crucially, it does not mean a run of tails is "due" to reverse; that belief is the gambler's fallacy.
Reading this result: With a near-fair coin the running proportion wanders wildly at first, but the law pulls it toward 0.50 as flips pile up — and a streak of tails never makes heads “due.”
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How it works
As the number of trials grows, the observed average converges to the true expected value — the foundation of statistics, insurance, and casinos. It is often misread as implying a run of tails makes heads "due," but each flip stays independent; that belief is the gambler's fallacy. Convergence comes from swamping early noise with sheer volume, not from any self-correction.
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The solver uses established numerical methods, but results are for research and educational purposes and should be validated against experiment or professional review before you rely on them.