The number of red balls and white balls in an urn is unknown, but the proportion, $π$, of red balls is either $1/3$ or $1/2$. A random sample of size $5$, drawn with replacement, yields the sequence: red, white, white, red and white. What is the most likely value of $\pi$?
I know how to solve this question , but I am wondering why we can't estimate $\pi$ using sample mean, since getting a red ball or not could be considered a Bernoulli distribution, which has a suitable estimator of $X$ bar. Is it because number of balls unknown?
Let $E$ be the observed event. The sample mean would be the answer to $$ \sup_{0\leq\pi\leq1}P(E) $$ whereas in our case we want the solution to $$ \max_{\pi\in\{1/3, 2/3\}} P(E). $$