I am trying to solve this problem:
Consider tossing a fair coin 100n times, where n is geometrically distributed with mean 6. Due to a counting error, it is reported that the fraction of heads is in [0.5, 0.51]. What is the most likely value for n?
I know I have to use the central limit theorem, which basically says that all distributions become normal as $n \rightarrow \infty$ but the part about the number of tosses itself being a random variable is confusing me. I am wondering if I have to use conditional probability/bayes rule where I condition n on the fraction of heads that I have observed ([0.5, 0.51]).
Central limit theorem: $P(a \leq \frac {S_n - n\mu} {\sqrt n \sigma} \leq b) \rightarrow \phi(b)-\phi(a)$ where $\phi$ is the integral of the error function.