Exactly half heads

Idea. The number of heads is close to a normal with mean 50 and standard deviation 5. The probability of one value is about the normal density at its centre.

Step 1: the normal approximation. With $\sigma =5$, the normal density at the mean is $\frac{1}{\sigma \sqrt{2\pi }}$, and a single integer value occupies a width of 1. So $$\mathbb{P}(50\text{ heads})\approx \frac{1}{5\sqrt{2\pi }}=\frac{1}{\sqrt{50\pi }}\approx \mathrm{0.0798.}$$

Step 2: the same with Stirling. $n!\approx \sqrt{2\pi n}(n/e{)}^n$ gives $\left(\genfrac{}{}{0ex}{}{2m}{m}\right)\approx \frac{4^m}{\sqrt{\pi m}}$, so with $m=50$ the probability is about $\frac{1}{\sqrt{50\pi }}$ again.

Step 3: how good is it? The exact value is $\left(\genfrac{}{}{0ex}{}{100}{50}\right)/2^{100}\approx 0.0796$. The approximation is off by about $0.0002$, a relative error near $\frac{1}{8m}=0.25\%$.

The point. Even the single most likely count happens less than 8% of the time. The chance of exactly half heads in $2m$ tosses falls like $\frac{1}{\sqrt{\pi m}}$.