As fair coin flips accumulate, does the gap between the head count and tail count shrink?

## Two quantities, two different fates Flip a fair coin `n` times. Let `H` be the number of heads and `T = n - H` the number of tails. Two natural summaries behave in opposite ways: - **The proportion** `H / n` converges to 0.5. This is the law of large numbers applied to indicator variables: each flip contributes 1 for heads and 0 for tails, the expected value of one flip is 0.5, and the running average approaches it. - **The difference** `D = |H - T| = |2H - n|` does *not* shrink. Its typical magnitude grows without bound as `n` grows, on the order of the square root of `n`. Both statements are true simultaneously, and holding them together is the whole lesson. ## Why they are compatible Write the proportion's deviation in terms of the difference: ``` H/n - 0.5 = (H - T) / (2n) = D_signed / (2n) ``` The numerator grows, but only like the square root of `n`; the denominator grows like `n` itself. A quantity growing like the square root of `n`, divided by `n`, goes to zero. So the proportion converges *because* the gap grows more slowly than the number of flips, not because the gap disappears. Concretely, at `n = 100` a gap of about 8 is typical; at `n = 10,000` a gap of about 80 is typical; at `n = 1,000,000` a gap of about 800 is typical. In absolute terms you are drifting steadily further from perfect balance. In relative terms — 8/100, 80/10,000, 800/1,000,000 — you are marching toward zero. ## Why people expect the opposite The intuition that the counts should equalise is the gambler's fallacy in a respectable coat. If the law of large numbers worked by compensation, the running gap would be actively pulled back to zero, and the counts would converge on each other. It does not work that way. Each flip is drawn afresh from the same distribution, and the running difference behaves like a wandering path with no restoring force: at each step it is equally likely to move up or down by one, so it diffuses away from its starting point. ## A famous nuance The running difference does return to zero. For a fair coin, the head and tail counts are exactly equal infinitely often with probability one — the wandering path keeps crossing its starting line forever. But the waiting times between successive ties get enormously long, and between ties the path can spend vast stretches entirely on one side. So "the counts are exactly equal infinitely often" and "the gap typically grows" are both correct: returns to balance are certain, but rarer and rarer, and the excursions between them get larger. Neither fact gives anyone a reason to bet on the next flip. ## Why this matters beyond coins The same split applies to any running total versus running average: - A rate — conversion rate, defect rate, click-through rate — stabilises with volume, and estimating it well is exactly what the law of large numbers licenses. - The corresponding raw *count* difference — how many more conversions than the expected number you have accumulated — does not stabilise. It drifts, and it drifts more in absolute terms the longer you run. That matters for anyone reading cumulative charts. A cumulative-difference line that keeps wandering away from zero is exactly what a fair, no-effect process looks like; it is not evidence of a trend. Plotting the ratio rather than the running total is usually what you actually want, precisely because the ratio is the quantity the law of large numbers promises will settle. ## Answering it well Say no, then give both halves in one breath: the proportion converges, the absolute gap typically grows like the square root of the number of flips, and the two are compatible because the gap grows more slowly than the count. If you add that the counts nonetheless hit exact equality infinitely often, you have shown you know the difference between a limit statement about ratios and a claim about balance.