Sorry. This is about maths and stats, with a possible application to physical geography. Nothing to do with economics (well, it might have an application to business cycles, but I'm not ready to go there yet).
Maybe the geographers have already figured this one out long ago (I couldn't find it on Google, but I was probably asking the wrong question).
This question has been stuck in my brain for months. I can't answer it. I can't even ask the question properly. And it's really bugging me that I can't ask it and answer it properly. But I think it can be asked properly, so someone could solve it with basic math and stats.
Wikipedia tells me that 9% of Canada is covered by water. I thought it would be higher.
This is my attempt to state the problem:
Here are my simplifying assumptions:
Each country is an island in the ocean.
The elevation of each point in a country is random. But the elevation of any point would have to be positively correlated with the elevation of adjacent points otherwise countries would just be miniscule dots in the ocean.
Evaporation is very small relative to rainfall, and the surface is impermeable to water, so that every area that is lower than a contour line that surrounds it has a lake, and every lake has an outlet to the ocean where the excess rainfall spills out.
I think that the percentage of a country covered in lakes would be an increasing function of the surface area of a country, and a decreasing function of the average elevation of a country. So take the limit as "the surface area gets large relative to the average elevation". (I know those two things don't have the same units, so I can't be stating the problem correctly here.) I think that that limit exists. Call it P*.
My gut says that P*=50%. That's because U-shaped bits of the country will be covered by water, and inverse-U-shaped bits of the country will not be covered by water. Small countries (where surface area is small relative to the average elevation) will be inverse-U-shaped, but in the limit, as a country gets large, there should be as many U-shaped bits as inverse-U-shaped bits. So P*=50% looks like it might be the right answer. But I might also be very wrong.
This is my very incomplete attempt to solve the problem.
Get a big map of the country. Divide it into a grid. Each "point" on the grid is (say) one centimetre square. Each point is flat.
Each point is surrounded by 4 other adjacent points: to the North, South, East, and West. Except for points on the coast, which have the ocean on at least one side.Each point on the coast is dry (not covered in water). It has to be dry, because it has to have a higher elevation than sea level, the sea is adjacent, and so any rain falling on the coast runs downhill into the sea.
For any point not on the coast: each of the 4 points adjacent to a given point has a 50% probability of being at a higher (or lower) elevation than the given point, and those probabilities are independent of each other. That means that each point has a 1/8 probability of being surrounded by 4 points all higher than it is, which is a sufficient condition for that point being wet (covered in water).
If any of the 4 adjacent points is wet, and if that point is higher than the given point, that is a second sufficient condition for the given point being wet. Each adjacent point has a probability P* of being wet, and a probability P*/2 of being both wet and higher than the given point.
Then my brain got stuck.