Imagine I have a binomial distribution with n possible outcomes, where the ith outcome has a probability $p_i$ of occurring. If I am only allowed to sample one particular output, the kth output, and I measure this outcome N times I want to know the variance in my estimate of $p_k$. For example with 100 runs of the trial if I measure the kth output 10 times, I can guess $p_k=0.1$, but what is the variance of this estimate?
I know that if I generate a multinomial distribution with n elements the variance of the number of kth outputs is $np_k(1-p_k)$. Does this still hold in my situation, where I am only sampling a single output?
If you are only focusing on a particular outcome of a multinomial distribution, then the scenario reduces to the simple binomial setting, where a "success" is "getting the $k$th output" and "failure" is "getting any other output."
Specifically, $N \sim \text{Binomial}(n, p_k)$ where $p_k$ is the true unknown probability of the $k$th output, and $n$ is the number of trials. $N$ has variance $np_k(1-p_k)$, so your estimate $\hat{p}_k = N/n$ has variance $\frac{p_k(1-p_k)}{n}$.