You repeatedly toss a weighted coin 100 times. You know that the probability of getting heads, $p,$ is either 0.1 or 0.01 (the probability is the same across all tosses), but you don't know which. What is the decision boundary in terms of the number of times out of 100 that you get heads for the MLE of the parameter $p?$
My friend gave me this problem, but I feel like it's not well defined. Isn't the MLE here just going to be the number of heads divided by $100?$ Or am I missing something fundamental here?
You are now told that it is either $0.1$ or $0.01$. Even if you get exactly $50$ head, we know that $0.5$ is not the solution. That is rather than optimizing over $p \in [0,1]$, we should only maximize over $p \in \{0.1, 0.01\}$.
Here is how to approach the solution:
If $p = 0.1$, what is the likelihood that you observe $n$ heads. What about when $p=0.01$? Now compare which likelihood is higher.