The Universe of Discourse
Thu, 20 Jul 2006

Flipping coins, corrected
In a recent article about coin flipping, I said:

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


In general, if you flip the coin n times, the expected difference between the numbers of heads and tails will be about √n.

In fact, the expected difference is actually !!\sqrt{2n/\pi}!!. For n=1,000,000, this gives an expected difference of about 798, not 1,000 as I said.

I correctly remembered that the expected difference is on the order of √n, but forgot that the proportionality constant was not 1.

The main point of my article, however, is still correct. I said that the following assertion is not quite right (although not quite wrong either):

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

I pointed out that although the relative difference tends to get small, the absolute difference tends to infinity. This is still true.

Thanks to James Wetterau for pointing out my error.

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