The question looks pretty simple but I can't get my hands on it:
Say I have a probability which is the product of two other independent probabilities $p = p_1p_2$.
I have estimated each probability $p_1$ and $p_2$ and found some $95\%$ confidence interval for each. How do I obtain a $95\%$ confidence interval for $p$?
Taking the product of the bounds of the interval won't work as I would be taking $95\%$ of a $95\%$ confidence interval resulting in an approximately $90\%$ which is not what I want.
So conversely I would be taking $97.5\%$ confidence interval for each and by multiplying the bounds I will obtain a $95\%$ confidence interval, is that right?
I feel like something is going wrong. In my situation I deal with probabilities but it could be anything so this question can be generalised to any type of confidence intervals.
If my reasoning is correct, could someone convince me that's the correct way of doing so?