Suppose we have three categories with $p_1+p_2+p_3=1$. And suppose we have already derived these two (say 95%) symmetric confidence intervals: $$ p_1 \in\ (L_1, U_1),\quad p_1 + p_2 \in\ (L_2,U_2). $$
Can we derive the (95%) confidence interval for $p_2$ given only this information?
I suppose that a lower bound for the interval is greater than $L_2 - U_1$, since we are 95% confident that $p_1<U_1$, but is this an equality? Here is my thought process:
$$Pr(p_1<U_1)=0.975,\ Pr(p_1+p_2>L_2)=0.975.$$ $$Pr(p_2>L_2-U_1) = Pr(p_1+p_2>L_2 \text{ and } p_1<U_1)=0.975^2.$$
That last line I'm not 100% sure about, I guess I just need a little guidance in the right direction for how to think about this. Any such guidance is greatly appreciated!