Stirling Approximation

Example

$$ b(n, ~2n, ~p) \approx {(4pq) \over \sqrt{\pi n}} $$

where $n$ is the number of heads, $2n$ is the number of trials, and $p$ is the probability of success.

$$ b(n, ~2n, ~p) = {2n \choose n} p^n (1-p)^{2n-n} $$

$$ = {(2n)! \over {n!n!}} p^n q^n $$

By Stirling's approximation,

$$ \approx {{\sqrt{2\pi ~ 2n} ~ {2n}^{2n} ~ e^{-2n} ~ p^n ~ q^n} \over {\sqrt{2 \pi n} ~ n^n ~ e^{-n}}} $$

Cleaning up,

$$ = {(4pq) \over \sqrt{\pi n}} ~~~~~~~~~~ \blacksquare $$