issue with binomial distribution

Question

The PMF of binomial distribution is $$B_k(N,p)=\frac{N!}{k!(N-k)!}p^k(1-p)^{N-k}.$$ If we have $p=1/2$ and $k=N/2$, is it is permissible to take the limit $N\rightarrow\infty$? Becouse if we do that $$\lim_{N\rightarrow\infty}B_{N/2}(N,1/2)=0.$$ As if one could not find half heads and half tails with a large N of coins.

Answer 1

It is permissible, and you are right that the limit is zero. This reflects the fact that the chance of getting exactly half heads goes to zero as $N \to \infty$. You are very likely to be close to half heads, but not exactly.