Maximum Likelihood for a Binomial

Key idea. The binomial coefficient does not depend on $p$, so it is enough to maximise $\log\bigl(p^8(1-p)^4\bigr)$.

We have

and $\binom{12}{8}$ is a positive constant. The logarithm is strictly increasing, so we may maximise

Differentiating,

The denominator is positive on $(0,1)$. Hence $\ell'(p)>0$ for $p<\tfrac23$ and $\ell'(p)<0$ for $p>\tfrac23$.

So $\ell$ increases up to $p=\tfrac23$ and decreases afterwards. The point $p=\tfrac23$ is therefore the unique global maximiser on $(0,1)$. At this point

Generalisation. For $X\sim\mathrm{Bin}(n,p)$ and an observation $0<k<n$, the same computation gives

The maximum likelihood estimate of $p$ is therefore the sample proportion $\hat p=k/n$.