Binomial distribution variance of the estimator

Question

this might be a duplicate, so I apologize in advance. Based on a 10000 Bernoulli trials I estimated that the probability of getting 1 is 57%. My professor told me I should also calculate the variability of my result and that is where I got confused. I know that the variability of MLE of parameter $p$ in the binomial distribution is $p(1-p)/n$, if I multiply it by 100 I get the result as a percentage. However, I don't know what $p$ is. Should I use the MLE estimator of the variance? That is $0.57*0.43*100/10000=0.002$%. Or am I missing something? How would you proceed? Thank you.

Answer 1

Yes. Because the MLE $\hat p=\frac{\sum_{i=1}^{10000}X_i}{10000}$ it has variance $\frac{p(1-p)}{10000}$. Not knowing the true $p$, it is reasonable to use the MLE. // Since there are a lot of trials, the variability is low.