What is the maximum likelihood estimate of $p$?

Question

$1,2,3,3,4$ is a random sample from $X$ that has the following probability distribution:
$P(X=1)=P(X=2)=1/4$.
$P(X=3)=p$
$P(X=4)=\frac{1}{2}-p$
Find the Maximum likelihood estimate for $p$
I have solved many problems on finding MLE's and I know only the method of finding the likelihood function and differentiating it. I have no clue how to solve this one.

Answer 1

The joint prob is $(1/4)^2p^2(1/2 - p)$. Differentiate w.r.t $p$, set the derivative to $0$ to get $2p(1/2-p)-p^2=0$ i.e. $p-3p^2=0$ i.e. $p=1/3$

Answer 2

It is a multinomial distribution, i.e., the likelihood function is $$ \mathcal{L}(p;\mathrm{x} =( 1,2, 3, 3, 4) ) = \frac{5!}{1!1!2!1!}\frac{1}{4^2}p^2(1/2- p), $$ the log likelihood is $$ \ell(p) = \ln \left( \frac{5!}{1!1!2!1!}\frac{1}{4^2}\right) + 2\ln p+\ln(1/2-p), $$ $$ \ell'(p)|_{p= \hat{p}} = \frac{2}{p}- \frac{1}{1/2-p}=0 \to \hat{p}=1/3. $$ Checking the sign of second derivative $$ \ell''(p)|_{p= 1/3} = -\frac{2}{1/9} - \frac{1}{(1/2-1/3)^2}<0. $$