Sam flips a coin $n$ times in order to check if its fake or not.
A fake coin has probability of $\frac{1}{3}$ to get heads, while a real one has a probability of $\frac{1}{2}$ to get heads.
Let $p_n$ be the "frequency" of the outcome "head" after $n$ flips, if $p_n$ is closer to $\frac{1}{3}$, then Sam guesses that it's a fake coin, and if it's closer to $\frac{1}{2}$ Sam guesses that it's real coin.
Given that the coin is real, what is the minimal number $n$ of flips that Sam has to do in order to ensure with probability atleast $0.95$ that he will guess right (Use CLT)?
My thoughts:
Knowing that the coin is real, then we know that the expected value of each throw is $\frac{1}{2}$, and the variance is $\frac{1}{4}$.
Now using CLT, we can say that $p_n\sim N(\frac{1}{2}n, 0.25n)$.
And to make sure he guesses right, $p_n$ should be closer to $\frac{1}{3}$ in a probability that is less than $0.05$, which means:
$P(p_n < \frac{5}{12})<0.05$.
Here I stopped to take look at the hint as I've had my plan set, and what was written is $p_n\sim N(0.5,0.25\frac{1}{n})$ - according to CLT, I currently have no idea why this is true, how did they reach this conclusion from the question? and where did I misunderstand things in the question?
Any help is appreciated.