the maximum likelihood estimator of $p?$

Question

$X$ is a single observation from Binomial $(1, p)$ population, where $p \in [1/5, 4/5]$ is unknown. If the observed value of $X$ is zero, then the maximum likelihood estimator of $p?$

I am stuck with this problem, please help.

Answer 1

The pmf of a bernoulli random variable is $$ P(X=x)=p^x(1-p)^{1-x}; \quad (x=0,1)\tag{1}\label{1} $$ Given an observation $x=0$, the Likelihood function is $$ L(p)=1-p $$ by substituting $x=0$ into $\eqref{1}$. If $p\in[1/5, 4/5]$ what value of $p$ maximizes $L(p)$.