I was catching up on Engines of our Ingenuity episodes this evening when the following line jumped out at me:

If I flip a coin a million times, I’m virtually certain to get 50 percent heads and 50 percent tails.

Depending on how you understand that line, it’s either imprecise or false. The more times you flip the coin, the **more** likely you are to get **nearly **half heads and half tails, but the **less** likely you are to get **exactly** half of each. I assume Dr. Lienhard knows this and that by “50 percent” he meant “nearly half.”

Let’s make the fuzzy statements above more quantitative. Suppose we flip a coin 2*n* times for some large number *n*. Then a calculation using Stirling’s approximation shows that the probability of *n* heads and *n* tails is approximately

1/√(π*n*)

which goes to zero as *n* goes to infinity. If you flip a coin a million times, there’s less than one chance in a thousand that you’d get exactly half heads.

Next, let’s quantify the statement that *nearly* half the tosses are likely to be heads. The normal approximation to the binomial tells us that for large *n*, the number of heads out of 2*n* tosses is approximately distributed like a normal distribution with the same mean and variance, i.e. mean *n* and variance *n*/2. The *proportion* of heads is thus approximately normal with mean 1/2 and variance 1/8*n*. This means the standard deviation is 1/√(8*n*). So, for example, about 95% of the time the proportion of heads will be 1/2 plus or minus 2/√(8*n*). As *n* goes to infinity, the width of this interval goes to 0. Alternatively, we could pick some fixed interval around 1/2 and show that the probability of the proportion of heads being outside that interval goes to 0.

“As n goes to zero, the width of this interval goes to 0.”

The first “zero” should be “infinity,” no?