On the sum of random coin flips

Question

If I toss a (fair) coin $N$ times, I expect to see an excess of tails/heads by roughly $\sqrt{N}$. Is this true? If yes, according to what theorem? If no, is there a way to quantify a bound for such possible excess?

(linked)

Answer 1

Number of heads when tossing a coin $N$ times is a binomial random variable, $\sim B(N,p)$. The variance of a binomial is $Npq, q=1-p$. Given a coin is fair, $p=q=1/2$, the expected value of (absolute) difference of heads and tails is 2 standard deviations, it is $$2 \sqrt{N/4}=\sqrt{N}$$

So you are correct.