The Universe of Discourse

Flipping coins
A gentleman on IRC recently said:

Over a million tosses you'll have almost the same amount of heads as tails

Well, yes, and no. It depends on how you look at it.

After a million tosses of a fair coin, you can expect that the numbers of heads and tails will differ by about 1,000. This is a pretty big number.

On the other hand, after a million tosses of a fair coin, you can expect that the numbers of heads and tails will differ by about 0.1%. This is a pretty small number.

In general, if you flip the coin n times, the expected difference between the numbers of heads and tails will be about √n. As n gets larger, so does √n. So the more times you flip the coin, the larger the expected difference in the two totals.

But the relative difference is the quotient of the difference and the total number of flips; that is, √n/n = 1/√n. As n gets larger, 1/√n goes to zero. So the more times you flip the coin, the smaller the expected difference in the two totals.

It's not quite right to say that you will have "almost the same amount of heads as tails". But it's not quite wrong either. As you flip the coin more and more, you can expect the totals to get farther and farther apart—but the difference between them will be less and less significant, compared with the totals themselves.

[ Addendum 20060720: Although the main point of this article is correct, I made some specific technical errors. A correction is available. ]

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