Approximate the probability of getting 500 heads out of a 1000 coin flip of unbiased coins to be within 5% of its true value (without the use of a calculator).
I know that an exact probability is $$\binom{1000}{500}(.5)^{1000} = .02522...$$
I am unsure how one could simplify this problem through estimation to get an approximate answer however.
Thanks for any help.
To avoid a calculator, you certainly need Stirling's approximation for the factorials. So $P=\binom {1000}{500}2^{-1000} \approx \frac {1000^{1000}\exp(500)\exp(500)}{500^{500}500^{500}\exp(1000)}\frac {\sqrt{2\pi 1000}}{\sqrt{2\pi 500}\sqrt{2 \pi 500}}2^{-1000}=\frac 1{\sqrt{\pi 500}}\approx \frac 1{\sqrt{1550}} \approx \frac 1{40}=0.025$
Stirling's approximation is within a factor $\frac 1{12n}$, so the error is negligible. Using $\pi \approx 3.1$ is within 2%. The last we were within 4% under the square root sign, so the root is within 2%, which means the calculation is within 4%. In fact this is within 1% of your exact value.