[gus_massa]

The surface of Earth has 5E8 km². They have 1E4 satellites. They are in orbit for a few years, but let's assume all fall down. So you get one satellite per 5E4 km².

There are less than 1E10 humans. 1m² is too big for most humans, perhaps a human and an umbrella. So the surface of humans is less than 1E10 m² = 1E4 km².

With some mild assumptions, I get less that a 20% chance of a satellite hitting a human, assuming all fall down, they can't be aimed at the sea, humans are big, they don't disintegrate in air, and the hit kills the human. How do they get 61%?

[Arnt]

The chance that one or more satellites will hit a human 1-x, where x is the chance that none of them will kill a human. x, in turn, is y¹⁰⁰⁰⁰, and y is the fraction of the earth's area where no human is. y¹⁰⁰⁰⁰ because it's the chance that the first satellite doesn't times the chance that the second times...

Raising something to the power of ten thousand makes a difference, even if you start very close to 1. I'm too tired to do the math now, but I bet that's where they get the big number and you don't.

[gus_massa]

Assuming my numbers are correct...

The chance that one satellite hits a (big) human is 1E4/5E8.

The chance that one satellite does not hits a (big) human is 1-1E4/5E8.

If you have 1E4 falling satellites, the chance that none of them hits a human is (1-1E4/5E8)^1E4.

If you have 1E4 falling satellites, the chance that at least one of them hits a human is 1-(1-1E4/5E8)^1E4.

https://www.wolframalpha.com/input?i=1-%281-1E4%2F5E8%29%5E%...

18.127% but my numbers are too rounded so I prefer to round the result to 20% (or at least 18%).

When you have (1-1/A)^B there are two different approximations.

When B > A you can approximate it by e^(-B/A)

When B < A you can approximate it by 1-B/A

The idea is that you can apply the Taylor serie and get (1-1/A)^B = e^(log(1-1/A)B) ~= e^((-1/A-1/A^2)B) ~= e^(-B/A-B/A^2) ~= 1-B/A-B/A^2+(-B/A-B/A^2)^2/2 ~= 1-B/A-B/A^2+B^2/A^2/2 ~= 1-B/A+B^2/A^2/2 dropping that tiny -B/A^2

So the final result of the approximation is

1-(1-B/A+B^2/A^2/2) = B/A - B^2/A^2/2

that in my case is

1E4/5E81E4 - (1E4/5E81E4)^2/2 = .2 - .04/2 = 20%-2% = 18%

The thing is that the first part of the approximation B/A is easy to remember. And the first correction is (B/A)^2*something, but I never remember that something is just 1/2, I must redo the calculation when I need it. Usually B/A is fine.

So ... I still don't understand if I made an horrible mistake with my selection of the numbers to make the approximation of they are doing something wrong.