The bell curve/gendarme’s hat is tall in the middle and very flat near the edges, which is to say that the farther a discrepancy is from zero, the less likely it is to be encountered. And this can be precisely quantified. If you flip N coins, the chance that you’ll end up being off by at most the square root of N from 50% heads is about 95.45%. The square root of 1,000 is about 31; indeed, eighteen of our twenty big thousand-coin trials above, or 90%, were within 31 heads of 500. If I kept playing the game, the fraction of times I ended up somewhere between 469 and 531 heads would get closer
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