| 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. |