How Not to Be Wrong: The Power of Mathematical Thinking

by Jordan Ellenberg

As you can see, the fraction of heads converges inexorably toward 50% as you flip more and more coins, as if squeezed by an invisible vise. You can see the same effect in the simulations. The proportions of heads in the first group of tries, the Smalls, range from 30% to 90%. With a hundred flips at a time, the range narrows: just 40% to 60%. And with a thousand flips, the range of proportions is only 46.2% to 53.7%. Something is pushing those numbers closer and closer to 50%. That something is the cold, strong hand of the Law of Large Numbers. I won’t state that theorem precisely (though it is stunningly handsome!), but you can think of it as saying the following: the more coins you flip, the more and more extravagantly unlikely it is that you’ll get 80% heads. In fact, if you flip enough coins, there’s only the barest chance of getting as many as 51%!