It’s well known that the population of Japan has been decreasing for years, and so I was a little puzzled by a recent headline saying that Japan’s population has dropped in every one of its 47 prefectures. Although the national population is in decline, until now not all of the nation’s 47 prefectures dropped in population at the same time. Is this remarkable? Actually, yes.

This post will not be about Japan’s demographics in detail, but was motivated by the headline mentioned above.

Demographics is not random; it is the result of individual human choices. But for this post we’ll look at demographics as if it were random.

Suppose each prefecture had the same probability *p* of declining. Then we could infer that *p* is probably quite large for all prefectures to decline at the same time. If you flip a coin 47 times and it comes up heads each time, you don’t imagine it’s equally likely to come up heads or tails the next time. (More on that here.)

For a Bernoulli random variable, how large would *p* have to be in order for the probability of 47 successes out of 47 trials to have probability at least 1/2?

If *p*^{47} = 1/2, then *p* = (1/2)^{1/47} = 0.985.

If the value of *p* is different in each prefecture, and the product of all the *p*‘s is 1/2, then all the *p* must be at least 1/2, and nearly all much larger. If one of the *p*‘s were equal to 1/2, the rest would have to be 1.

Applying this understanding to the headline at the top of the post, we could say that the fact that every prefecture has gone down in population is strong evidence that population will continue to drop, as long as current conditions continue.