It’s probably worth figuring out what went wrong in Approach 1 to Example 1, which I think is this part:
[300 cities of 10,000 or more people per county] × [2500 counties in the USA]
Note that this gives 750,000 cities of 10,000 or more people in the US, for a total of at least 7.5 billion people in the US. So it’s already clearly wrong here. I’d say 300 cities of 10,000 people or more per county is way too high; I’d put it at more like 1 (Edit: note that this gives at least 250 million people in the US and that’s about right). This brings down the final estimate from this approach by a factor of 300, or down to 3 million, which is much closer.
(Verification: I just picked a random US state and a random county in it from Wikipedia and got Bartow County, Georgia, which has a population of 100,000. That means it has at most 10 cities with 10,000 or more people, and going through the list of cities it actually looks like it only has one such city.)
This gives about 2,500 cities in the US total with population 10,000 or more. I can’t verify this number, but according to Wikipedia there are about 300 cities in the US with population 100,000 or more. Assuming the populations of cities are power-law distributed with exponent 1, this means that the nth-ranked city has population about 30,000,000/n, so this gives about 3,000 cities in the US with population 10,000 or more.
And in fact we didn’t even need to use Wikipedia! Just assuming that the population of cities is power-law distributed with exponent 1, we see that the distribution is determined by the population of the most populous city. Let’s take this to be 20 million people (the number you used for New York City). Then the nth-ranked city in the US has population about 20,000,000/n, so there are about 2,000 cities with population 10,000 or more.
Edit: Found the actual number. According to the U.S. Census Bureau, as of 2008, the actual number is about 2,900 cities.
Incidentally, this shows another use of Fermi estimates: if you get one that’s obviously wrong, you’ve discovered an opportunity to fix some aspect of your model of the world.
It’s probably worth figuring out what went wrong in Approach 1 to Example 1, which I think is this part:
Note that this gives 750,000 cities of 10,000 or more people in the US, for a total of at least 7.5 billion people in the US. So it’s already clearly wrong here. I’d say 300 cities of 10,000 people or more per county is way too high; I’d put it at more like 1 (Edit: note that this gives at least 250 million people in the US and that’s about right). This brings down the final estimate from this approach by a factor of 300, or down to 3 million, which is much closer.
(Verification: I just picked a random US state and a random county in it from Wikipedia and got Bartow County, Georgia, which has a population of 100,000. That means it has at most 10 cities with 10,000 or more people, and going through the list of cities it actually looks like it only has one such city.)
This gives about 2,500 cities in the US total with population 10,000 or more. I can’t verify this number, but according to Wikipedia there are about 300 cities in the US with population 100,000 or more. Assuming the populations of cities are power-law distributed with exponent 1, this means that the nth-ranked city has population about 30,000,000/n, so this gives about 3,000 cities in the US with population 10,000 or more.
And in fact we didn’t even need to use Wikipedia! Just assuming that the population of cities is power-law distributed with exponent 1, we see that the distribution is determined by the population of the most populous city. Let’s take this to be 20 million people (the number you used for New York City). Then the nth-ranked city in the US has population about 20,000,000/n, so there are about 2,000 cities with population 10,000 or more.
Edit: Found the actual number. According to the U.S. Census Bureau, as of 2008, the actual number is about 2,900 cities.
Incidentally, this shows another use of Fermi estimates: if you get one that’s obviously wrong, you’ve discovered an opportunity to fix some aspect of your model of the world.