US warns Starlink satellites will start killing people
the-sun.com
the-sun.com
FFS this click-bait shit is killing me. Starlink is only an example, and likely less likely then all the other low orbit shit flying around up there.
Must the media be so sloppy and shameless?
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%?
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.
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.