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by Arnt
986 days ago
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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. |
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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.