Law of large numbers

For independent trials from the same distribution with mean $\mu$ (mu), the sample mean $\bar{x}$ (x-bar) of the first $n$ results gets close to $\mu$ and stays close as $n$ grows. A proportion is the case where every result is a 0 or a 1, so the fraction of heads settles toward 0.5 for a fair coin. The claim is about the average. It says nothing whatever about the running count.

That distinction is the entire entry, so work it. Suppose the first 100 flips of a fair coin give 60 heads, a surplus of 10 above the expected 50. Keep flipping. The remaining flips are still fair, so the expected number of heads in the next 900 is 450, and the expected total at 1,000 flips is 510, still exactly 10 above half. At 10,000 flips it is 5,010. At 1,000,000 flips it is 500,010. The expected surplus never shrinks by a single flip.

The proportion converges anyway: 0.600, then 0.510, then 0.5010, then 0.50001. Those 10 extra heads are being divided by a denominator that grows without bound, so they vanish as a fraction while surviving intact as a count. Nothing repays them.

So tails is due after a run of heads is false, and it is false for a stateable reason: the coin has no memory and the law never promised a correction. That is the gambler's fallacy. The typical distance of the count from half actually grows, because the standard deviation of the number of heads in $n$ flips is $0.5\sqrt{n}$, which is 5 at 100 flips, 50 at 10,000, and 500 at 1,000,000. The standard deviation of the proportion is $0.5/\sqrt{n}$ over the same range, falling from 0.05 to 0.005 to 0.0005.

The conditions carry weight. A process whose probability drifts between trials, or draws taken without replacement from a small population, is not what the statement above covers. Topic 2.3 uses the law to justify why more simulated trials give a better estimate.

This is the reason a physics lab asks for repeated trials. Averaging many measurements drives the result toward the true value, which is what makes repetition a method of reducing uncertainty rather than just extra work: experimental uncertainty.