Maximum Likelihood Estimator of $\theta$

Question

I have the following question

I tried to answer

I got answer that same like this answer

Is this true answer? (Note that: in the question $0<p<\frac{1}{2}$, but in this answer $0<p<1$ ) Is this an effect observed in the solution?

Answer 1

No, it is not. You are maximizing the function on $[0,1]$, while you should maximize it on $[0,\frac{1}{2}]$. From the definition of MLE $\hat{p}(x)=\arg\max \{L(x,p):p \in[0,\frac{1}{2}] \}$.
The correct answer is $\bar{p}=\min\{\bar{x} ,\frac{1}{2} \}$.
Also note that, if initially $p \in (0,\frac{1}{2})$, then maximum likehood estimator wouldn't exist.