Re:Questions People Ask When I Tell Them I Own an Airplane
by vilhelm_s on Thursday February 20, 2014 @08:17PM
(#7274847)
That's why I said "approximately". :)
By the binomial series [1], the true value is 1-(1-p)^n = np - n(n-1)/2! p^2 + n(n-1)(n-2)/3! p^3 + ...
So approximating it to just np corresponds to taking just the leading term of the series. Of course, this is only valid when p is small enough. To see how good the approximation is, we can compare it to the two first terms of the series, which is np - n(n-1)/2 p^2, or approximately np - (np)^2 /2.
So the approximation np is about twice the percentage difference between the approximation and the true value. In this case, the approximation np is about 2%, and that guess is itself about 1% wrong.
By the binomial series [1], the true value is 1-(1-p)^n = np - n(n-1)/2! p^2 + n(n-1)(n-2)/3! p^3 + ...
So approximating it to just np corresponds to taking just the leading term of the series. Of course, this is only valid when p is small enough. To see how good the approximation is, we can compare it to the two first terms of the series, which is np - n(n-1)/2 p^2, or approximately np - (np)^2 /2.
So the approximation np is about twice the percentage difference between the approximation and the true value. In this case, the approximation np is about 2%, and that guess is itself about 1% wrong.
[1] https://en.wikipedia.org/wiki/Binomial_series